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*Table of contents : CoverTitle PageCopyright PagePrefaceAcknowledgmentsTable of ContentsSection I: The Generalized Method of Lines 1. The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation 1.1 Introduction 1.2 On the numerical procedures for Ginzburg-Landau type ODEs 1.3 Numerical results for related P.D.E.s 1.3.1 A related P.D.E on a special class of domains 1.3.2 About the matrix version of G.M.O.L. 1.3.3 Numerical results for the concerning partial differential equation 1.4 A numerical example concerning a Ginzburg-Landau type equation 1.4.1 About the concerning improvement for the generalized method of lines 1.4.2 Software in Mathematica for solving such an equation 1.5 The generalized method of lines for a more general domain 1.6 Conclusion 2. An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines 2.1 Introduction 2.2 The numerical method 2.3 A numerical example 2.4 A general proximal explicit approach 2.5 Conclusion 3. Approximate Numerical Procedures for the Navier-Stokes System through the Generalized Method of Lines 3.1 Introduction 3.2 Details about an equivalent elliptic system 3.3 An approximate proximal approach 3.4 A software in MATHEMATICA related to the previous algorithm 3.5 The software and numerical results for a more specific example 3.6 Numerical results through the original conception of the generalized method of lines for the Navier-Stokes system 3.7 Conclusion 4. An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model 4.1 Introduction 4.2 Applications to a flight mechanics model 4.3 AcknowledgementsSection II: Calculus of Variations, Convex Analysis and Restricted Optimization 5. Basic Topics on the Calculus of Variations 5.1 Banach spaces 5.2 The Gâteaux variation 5.3 Minimization of convex functionals 5.4 Sufficient conditions of optimality for the convex case 5.5 Natural conditions, problems with free extremals 5.6 The du Bois-Reymond lemma 5.7 Calculus of variations, the case of scalar functions on Rn 5.8 The second Gâteaux variation 5.9 First order necessary conditions for a local minimum 5.10 Continuous functionals 5.11 The Gâteaux variation, the formal proof of its formula 6. More Topics on the Calculus of Variations 6.1 Introductory remarks 6.2 The Gâteaux variation, a more general case 6.3 Fréchet differentiability 6.4 The Legendre-Hadamard condition 6.5 The Weierstrass condition for n = 1 6.6 The Weierstrass condition, the general case 6.7 The Weierstrass-Erdmann conditions 6.8 Natural boundary conditions 7. Convex Analysis and Duality Theory 7.1 Convex sets and functions 7.2 Weak lower semi-continuity 7.3 Polar functionals and related topics on convex analysis 7.4 The Legendre transform and the Legendre functional 7.5 Duality in convex optimization 7.6 The min-max theorem 7.7 Relaxation for the scalar case 7.8 Duality suitable for the vectorial case 7.8.1 The Ekeland variational principle 7.9 Some examples of duality theory in convex and non-convex analysis 8. Constrained Variational Optimization 8.1 Basic concepts 8.2 Duality 8.3 The Lagrange multiplier theorem 8.4 Some examples concerning inequality constraints 8.5 The Lagrange multiplier theorem for equality and inequality constraints 8.6 Second order necessary conditions 8.7 On the Banach fixed point theorem 8.8 Sensitivity analysis 8.8.1 Introduction 8.9 The implicit function theorem 8.9.1 The main results about Gâteaux differentiability 9. On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces 9.1 Introduction 9.2 The Lagrange multiplier theorem for equality constraints and non-smooth optimization 9.3 The Lagrange multiplier theorem for equality and inequality constraints for non-smooth optimization 9.4 ConclusionSection III: Duality Principles and Related Numerical Examples Through the Generalized Method of Lines 10. A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization 10.1 Introduction 10.2 The main duality principle, a convex dual variational formulation 11. Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations 11.1 Introduction 11.2 A general duality principle non-convex optimization 11.3 Another duality principle for a simpler related model in phase transition with a respective numerical example 11.4 A convex dual variational formulation for a third similar model 11.4.1 The algorithm through which we have obtained the numerical results 11.5 An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation 11.6 An exact convex dual variational formulation for a non-convex primal one 11.7 Another primal dual formulation for a related model 11.8 A third primal dual formulation for a related model 11.9 A fourth primal dual formulation for a related model 11.10 One more primal dual formulation for a related model 11.11 Another primal dual formulation for a related model 12. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations 12.1 Introduction 12.2 The main duality principle, a convex dual formulation and the concerning proximal primal functional 12.3 A primal dual variational formulation 12.4 One more duality principle and a concerning primal dual variational formulation 12.4.1 Introduction 12.4.2 The main duality principle and a related primal dual variational formulation 12.5 One more dual variational formulation 12.6 Another dual variational formulation 12.7 A related numerical computation through the generalized method of lines 12.7.1 About a concerning improvement for the generalized method of lines 12.7.2 Software in Mathematica for solving such an equation 12.7.3 Some plots concerning the numerical results 12.8 Conclusion 13. A Note on the Korn’s Inequality in a N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model 13.1 Introduction 13.2 The main results, the Korn inequalities 13.3 An existence result for a non-linear model of plates 13.4 On the existence of a global minimizer 13.5 ConclusionReferencesIndex*

The Numerical Method of Lines and Duality Principles Applied to Models in Physics and Engineering Fabio Silva Botelho

Federal University of Santa Catarina – UFSC Department of Mathematics, Campus Trindade Florianopolis – SC – Brazil

A SCIENCE PUBLISHERS BOOK

First edition published 2024 by CRC Press 2385 NW Executive Center Drive, Suite 320, Boca Raton FL 33431 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2024 Fabio Silva Botelho CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data (applied for)

ISBN: 978-1-032-19209-3 (hbk) ISBN: 978-1-032-19210-9 (pbk) ISBN: 978-1-003-25813-1 (ebk) DOI: 10.1201/9781003258131 Typeset in Times New Roman by Radiant Productions

Preface The first part of this text develops theoretical and numerical results concerning the original conception of the generalized method of lines as well as more recent advances relating to such a method. We recall that, for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or more generally, in curves), and the concerning solution is written on these lines as functions of the boundary conditions and the domain boundary shape. Beyond some improvements related to its original conception, we develop an approximate proximal approach that has been successful for a large class of models in physics in engineering, including results also suitable for a large class of domain shapes and concerning boundary conditions. In its first chapter, the text presents applications to a Ginzburg-Landau type equation. The results include an initial description of the numerical procedures and, in a second step, we present the related software either in MATHEMATICA or MATLAB® and examples of the corresponding line expressions for each problem considered. In the third chapter, we develop numerical results for the time independent, incompressible Navier-Stokes system in fluid mechanics. The results presented include a description of the numerical method in question, the corresponding software and the related line expressions obtained for each case considered. Furthermore, the establishment of an equivalent elliptic system is also presented. Here we highlight the procedure to obtain such a system is rather standard and well known, however, we emphasise the novelty is the identification of the correct concerning boundary conditions that lead to the mentioned equivalence between these two mentioned systems. In the second text part, we present basic topics on the calculus of variations, duality theory, and constrained optimization, including new results in Lagrange multipliers for non-smooth optimization in a Banach space context. Furthermore, a generalization of the Ekeland variational principle is also developed.

iv ■ The Method of Lines and Duality Principles for Non-Convex Models

Finally, in the third text part, we develop applications connecting the generalized method of lines and the duality theory. We recall that the dual and primal dual formulations are fundamentally important specially for systems that have no optimal global solution in a classical sense. In such a case, the critical points of the dual formulations correspond to weak cluster points of minimizing sequences for the primal ones, and reflect the average behavior of such sequences close to the corresponding global infima values for the primal models addressed. Such evaluations are important from both theoretical and practical points of view, since they may serve as a reliable tool for projects close to global optimality for a concerning primal model. At this point we start to describe a summary of each chapter content.

Summary of each book chapter content Chapter 1—The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation This chapter develops an improvement concerning the original format of the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves), and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. In this text, we introduce a slight change in the way we truncate the series solution generated through the application of the Banach fixed point theorem to obtain the relation between two adjacent lines. With such a new improvement, we have got very good results even as a typical parameter is too much small, decreasing substantially the numerical solution error for such a class of small parameters. In the last sections, we present numerical examples and results related to a Ginzburg-Landau type equation. Chapter 2—An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines This chapter develops an approximate proximal approach for the generalized method of lines. We recall again that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves), and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. Considering such a context, along the text we develop an approximate numerical procedure of proximal nature applicable to a large class of models in physics and engineering. Finally, in the last sections, we present numerical examples and results related to a Ginzburg-Landau type equation.

Preface ■ v

Chapter 3—Approximate Numerical Procedures for the Navier-Stokes System through the Generalized Method of Lines This chapter develops applications of the generalized method of lines to numerical solutions of the time-independent, incompressible Navier-Stokes system in fluid mechanics. Once more, we recall that for such a method, the domain of the partial differential equation in question is discretized in lines (or more generally in curves), and the concerning solutions are written on these lines as functions of the boundary conditions and the domain boundary shape. More specifically, in this text we present softwares and results for a concerning approximate proximal approach as well as results based on the original conception of the generalized method of lines. Chapter 4—An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model This chapter develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the generalized method of lines, originally published in the referenced books of 2011 and 2014 [22, 12], respectively. Finally, in the last section, we apply the method to a model in flight mechanics. Chapter 5—Basic Topics on the Calculus of Variations This chapter develops the main concepts on the basic calculus of the variations theory. Topics addressed include the Gâteaux variation, sufficient conditions for extremals of convex functionals, natural and essential boundary conditions, and the second Gâteaux variation. The chapter finishes with a formal proof of the Gâteaux variation formula for the scalar case. Chapter 6—More Topics on the Calculus of Variations In this chapter, we present some more advanced topics on the calculus of variations, including Fréchet differentiability, the Legendre-Hadamard condition, the Weierstrass condition, and the Weierstrass-Erdmann corner conditions. Chapter 7—Convex Analysis and Duality Theory This chapter develops basic and advanced concepts on convex analysis and duality theory. Topics such as convex functions, weak lower semi-continuity, polar functionals, and the Legendre transform are developed in detail. Among the topics on duality theory, we highlight the min-max theorem and the Ekeland variational principle.

vi ■ The Method of Lines and Duality Principles for Non-Convex Models

Chapter 8—Constrained Variational Optimization In this chapter, we introduce the main definitions and results on constrained variational optimization. Topics addressed include the duality theory for the convex case, the Lagrange multiplier theorem for equality, and equality/inequality restrictions in a Banach spaces context. Chapter 9—On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces This chapter develops optimality conditions for a large class of nonsmooth variational models. The main results are based on standard tools of functional analysis and calculus of variations. Firstly, we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach space context. We highlight, the results in general are well known, however, some novelties are introduced related to the proof procedures, which are in general softer than those concerning the present literature. Chapter 10—A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization This chapter develops a convex dual variational formulation for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis, and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. Chapter 11—Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations This chapter develops duality principles and numerical results for a large class of non-convex variational models. The main results are based on the fundamental tools of convex analysis, the duality theory, and calculus of variations. More specifically, the approach is established for a class of nonconvex functionals similar to those found in some models in phase transition. Finally, in the last section we present a concerning numerical example and the respective software. Chapter 12—Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations This chapter develops dual variational formulations for a large class of models in variational optimization. The results are established through the

Preface ■ vii

basic tools of functional analysis, convex analysis, and, the duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section, we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation. Chapter 13—A Note on the Korn’s Inequality in a n-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model In the first part of this chapter, we present a new proof for the Korn inequality in a n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result, the standard Poincaré inequality plays a fundamental role. In the second text part, we develop a global existence result for a non-linear model of plates. We address a rather general type of boundary conditions and the novelty here is the more relaxed restrictions concerning the external load magnitude.

Acknowledgments I would like to thank my colleagues and Professors from Virginia Tech - USA, where I got my Ph.D. degree in Mathematics in 2009. At Virginia Tech, I had the opportunity of working with many exceptionally qualified people. I am especially grateful to Professor Robert C. Rogers for his excellent work as advisor. I would like to thank the Department of Mathematics for its constant support and this opportunity of studying mathematics at an advanced level. Among many other Professors, I particularly thank Martin Day (Calculus of Variations), William Floyd and Peter Haskell (Elementary Real Analysis), James Thomson (Real Analysis), Peter Linnell (Abstract Algebra), and George Hagedorn (Functional Analysis) for the excellent lecture courses. Finally, special thanks to all my Professors at I.T.A. (Instituto Tecnológico de Aeronáutica, SP-Brasil) my undergraduate and masters school. Specifically about I.T.A., among many others, I would like to express my gratitude to Professors Leo H. Amaral, Tânia Rabelo, and my master thesis advisor Antônio Marmo de Oliveira, also for their valuable work. Fabio Silva Botelho December 2022 Florianópolis - SC, Brazil

Contents Preface

iii

Acknowledgments viii SECTION I: THE GENERALIZED METHOD OF LINES 1. The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation 1.1 Introduction 1.2 On the numerical procedures for Ginzburg-Landau type ODEs 1.3 Numerical results for related P.D.E.s 1.3.1 A related P.D.E on a special class of domains 1.3.2 About the matrix version of G.M.O.L. 1.3.3 Numerical results for the concerning partial differential equation 1.4 A numerical example concerning a Ginzburg-Landau type equation 1.4.1 About the concerning improvement for the generalized method of lines 1.4.2 Software in Mathematica for solving such an equation 1.5 The generalized method of lines for a more general domain 1.6 Conclusion 2. An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines 2.1 Introduction 2.2 The numerical method 2.3 A numerical example

2 2 3 5 5 7 9 10 11 14 17 25 26 26 27 30

x ■ The Method of Lines and Duality Principles for Non-Convex Models

2.4 A general proximal explicit approach 2.5 Conclusion

31 36

3. Approximate Numerical Procedures for the Navier-Stokes System 37 through the Generalized Method of Lines 3.1 Introduction 3.2 Details about an equivalent elliptic system 3.3 An approximate proximal approach 3.4 A software in MATHEMATICA related to the previous algorithm 3.5 The software and numerical results for a more specific example 3.6 Numerical results through the original conception of the generalized method of lines for the Navier-Stokes system 3.7 Conclusion

37 39 41 48

4. An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model

66

4.1 Introduction 4.2 Applications to a flight mechanics model 4.3 Acknowledgements

54 58 65

66 69 72

SECTION II: CALCULUS OF VARIATIONS, CONVEX ANALYSIS AND RESTRICTED OPTIMIZATION 5. Basic Topics on the Calculus of Variations

74

5.1 Banach spaces 74 5.2 The Gâteaux variation 81 5.3 Minimization of convex functionals 83 5.4 Sufficient conditions of optimality for the convex case 87 5.5 Natural conditions, problems with free extremals 89 5.6 The du Bois-Reymond lemma 96 5.7 Calculus of variations, the case of scalar functions on Rn 99 5.8 The second Gâteaux variation 101 5.9 First order necessary conditions for a local minimum 103 5.10 Continuous functionals 104 5.11 The Gâteaux variation, the formal proof of its formula 105 6. More Topics on the Calculus of Variations 6.1 Introductory remarks 6.2 The Gâteaux variation, a more general case

109 109 110

Contents ■ xi

6.3 6.4 6.5 6.6 6.7 6.8

Fréchet differentiability The Legendre-Hadamard condition The Weierstrass condition for n = 1 The Weierstrass condition, the general case The Weierstrass-Erdmann conditions Natural boundary conditions

7. Convex Analysis and Duality Theory 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Convex sets and functions Weak lower semi-continuity Polar functionals and related topics on convex analysis The Legendre transform and the Legendre functional Duality in convex optimization The min-max theorem Relaxation for the scalar case Duality suitable for the vectorial case 7.8.1 The Ekeland variational principle 7.9 Some examples of duality theory in convex and non-convex analysis 8. Constrained Variational Optimization 8.1 Basic concepts 8.2 Duality 8.3 The Lagrange multiplier theorem 8.4 Some examples concerning inequality constraints 8.5 The Lagrange multiplier theorem for equality and inequality constraints 8.6 Second order necessary conditions 8.7 On the Banach fixed point theorem 8.8 Sensitivity analysis 8.8.1 Introduction 8.9 The implicit function theorem 8.9.1 The main results about Gâteaux differentiability

112 112 114 116 120 123 126 126 126 129 132 136 139 147 155 155 161 167 167 171 172 178 179 183 186 188 188 188 193

9. On Lagrange Multiplier Theorems for Non-Smooth Optimization 200 for a Large Class of Variational Models in Banach Spaces 9.1 Introduction 9.2 The Lagrange multiplier theorem for equality constraints and non-smooth optimization 9.3 The Lagrange multiplier theorem for equality and inequality constraints for non-smooth optimization 9.4 Conclusion

200 204 206 210

xii ■ The Method of Lines and Duality Principles for Non-Convex Models

SECTION III: DUALITY PRINCIPLES AND RELATED NUMERICAL EXAMPLES THROUGH THE GENERALIZED METHOD OF LINES 10. A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization 10.1 Introduction 10.2 The main duality principle, a convex dual variational formulation 1 1. Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations 11.1 Introduction 11.2 A general duality principle non-convex optimization 11.3 Another duality principle for a simpler related model in phase transition with a respective numerical example 11.4 A convex dual variational formulation for a third similar model 11.4.1 The algorithm through which we have obtained the numerical results 11.5 An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation 11.6 An exact convex dual variational formulation for a non-convex primal one 11.7 Another primal dual formulation for a related model 11.8 A third primal dual formulation for a related model 11.9 A fourth primal dual formulation for a related model 11.10 One more primal dual formulation for a related model 11.11 Another primal dual formulation for a related model 12. Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations 12.1 Introduction 12.2 The main duality principle, a convex dual formulation and the concerning proximal primal functional 12.3 A primal dual variational formulation 12.4 One more duality principle and a concerning primal dual variational formulation 12.4.1 Introduction 12.4.2 The main duality principle and a related primal dual variational formulation 12.5 One more dual variational formulation

212 212 216 219 219 220 223 225 229 230 232 236 238 242 245 249 253 253 256 259 260 260 263 268

Contents ■ xiii

12.6 Another dual variational formulation 12.7 A related numerical computation through the generalized method of lines 12.7.1 About a concerning improvement for the generalized method of lines 12.7.2 Software in Mathematica for solving such an equation 12.7.3 Some plots concerning the numerical results 12.8 Conclusion 13. A Note on the Korn’s Inequality in a N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model 13.1 Introduction 13.2 The main results, the Korn inequalities 13.3 An existence result for a non-linear model of plates 13.4 On the existence of a global minimizer 13.5 Conclusion References

273 277 278 281 283 290 291 291 293 297 299 303 304

Index 311

THE GENERALIZED METHOD OF LINES

I

Chapter 1

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

1.1

Introduction

This chapter develops an improvement concerning the original format of the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves), and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. In this text, we introduce a slight change in the way we truncate the series solution generated through the application of the Banach fixed point theorem to obtain the relation between two adjacent lines. With such a new improvement, we have got very good results even though a typical parameter is very small, decreasing substantially the numerical solution error for such a class of small parameters. In the last sections, we present numerical examples and results related to a Ginzburg-Landau type equation. We start with a kind of matrix version of the Generalized Method of Lines. Applications are developed for models in physics and engineering.

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

1.2

■

3

On the numerical procedures for Ginzburg-Landau type ODEs

We first apply Newton’s method to a general class of ordinary differential equations. The solution here is obtained similarly to the generalized method of lines procedure. See the next sections for details on such a method for PDEs. For a C1 class function f and a continuous function g, consider the second order equation, ′′ u + f (u) + g = 0, in [0, 1] (1.1) u(0) = u0 , u(1) = u f . In finite differences we have the approximate equation: un+1 − 2un + un−1 + f (un )d 2 + gn d 2 = 0. Assuming such an equation is non-linear and linearizing it about a first solution {u}, ˜ we have (in fact, this is an approximation), un+1 − 2un + un−1 + f (u˜n )d 2 + f ′ (u˜n )(un − u˜n )d 2 + gn d 2 = 0. Thus we may write un+1 − 2un + un−1 + An un d 2 + Bn d 2 = 0, where, An = f ′ (u˜n ), and Bn = f (u˜n ) − f ′ (u˜n )u˜n + gn . In particular for n = 1 we get u2 − 2u1 + u0 + A1 u1 d 2 + B1 d 2 = 0. Solving such an equation for u1 , we get u1 = a1 u2 + b1 u0 + c1 , where a1 = (2 − A1 d 2 )−1 , b1 = a1 , c1 = a1 B1 . Reasoning inductively, having un−1 = an−1 un + bn−1 u0 + cn−1 , and un+1 − 2un + un−1 + An un d 2 + Bn d 2 = 0,

4

■

The Method of Lines and Duality Principles for Non-Convex Models

we get un+1 − 2un + an−1 un + bn−1 u0 + cn−1 + An un d 2 + Bn d 2 = 0, so that un = an un+1 + bn u0 + cn , where, an = (2 − an−1 − An d 2 )−1 , bn = an bn−1 , and cn = an (cn−1 + Bn d 2 ), ∀n ∈ 1, ..., N − 1. We have thus obtained un = an un+1 + bn u0 + cn ≡ Hn (un+1 ), ∀n ∈ {1, ..., N − 1}, and in particular, uN−1 = HN−1 (u f ), so that we may calculate, uN−2 = HN−2 (uN−1 ), uN−3 = HN−3 (uN−2 ), and so on, up to finding, u1 = H1 (u2 ). The next step is to replace {u˜n } by the {un } calculated, and repeat the process up to the satisfaction of an appropriate convergence criterion. We present numerical results for the equation, 3 u′′ − uε + εu + g = 0, in [0, 1] (1.2) u(0) = 0, u(1) = 0, where, and 1 g(x) = , ε The results are obtained for ε = 1.0, ε = 0.1, ε = 0.01 and ε = 0.001. Please see Figures 1.1, 1.2, 1.3 and 1.4 respectively.

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

■

5

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0 0

0.2

0.4

0.6

0.8

1

Figure 1.1: The solution u(x) by Newton’s method for ε = 1.

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Figure 1.2: The solution u(x) by Newton’s method for ε = 0.1.

1.3 1.3.1

Numerical results for related P.D.E.s A related P.D.E on a special class of domains

We start by describing a similar equation, but now in a two dimensional context. Let Ω ⊂ R2 be an open, bounded, connected set with a regular boundary denoted by ∂ Ω. Consider a real Ginzburg-Landau type equation (see [4], [9], [52], [53]

6

■

The Method of Lines and Duality Principles for Non-Convex Models

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Figure 1.3: The solution u(x) by Newton’s method for ε = 0.01.

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

Figure 1.4: The solution u(x) by Newton’s method for ε = 0.001.

for details about such an equation), given by ε∇2 u − αu3 + β u = f ,

u = 0,

in Ω (1.3) on ∂ Ω,

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

■

7

where α, β , ε > 0, u ∈ U = W01,2 (Ω), and f ∈ L2 (Ω). The corresponding primal variational formulation is represented by J : U → R, where J(u) =

1.3.2

Z

ε 2

∇u · ∇u dx +

Ω

α 4

Z

u4 dx −

Ω

β 2

Z

u2 dx +

Z

Ω

f u dx. Ω

About the matrix version of G.M.O.L.

The generalized method of lines was originally developed in [22]. In this work we address its matrix version. Consider the simpler case where Ω = [0, 1] × [0, 1]. We discretize the domain in x, that is, in N + 1 vertical lines obtaining the following equation in finite differences (see [71] for details about finite differences schemes). ε(un+1 − 2un + un−1 ) + εM2 un /d12 − αu3n + β un = fn , d2

(1.4)

∀n ∈ {1, ..., N − 1}, where d = 1/N and un corresponds to the solution on the line n. The idea is to apply the Newton’s method. Thus choosing a initial solution {(u0 )n } we linearize (1.4) about it, obtaining the linear equation: 2

un+1 − 2un + un−1

3αd (u0 )2n un + M˜2 un − ε β d2 d2 2α (u0 )3n d 2 + un − fn = 0, + ε ε ε

(1.5)

2

where M˜ 2 = M2 dd 2 and 1

M2 =

−2 1 0 0 1 −2 1 0 0 1 −2 1 .. .. .. .. . . . . 0 0 ··· 1 0 0 ··· ···

··· ··· ··· .. .

0 0 0 .. .

, −2 1 1 −2

(1.6)

with N1 lines corresponding to the discretization in the y axis. Furthermore d1 = 1/N1 . In particular for n = 1 we get 2

3αd (u0 )21 u1 u2 − 2u1 + M˜2 u1 − ε 2α β d2 d2 + (u0 )31 d 2 + u1 − f1 = 0. ε ε ε Denoting 2

2

βd αd M12 [1] = 2Id − M˜ 2 + 3 (u0 )21 Id − Id , ε ε

(1.7)

8

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The Method of Lines and Duality Principles for Non-Convex Models

where Id denotes the (N1 − 1) × (N1 − 1) identity matrix, Y0 [1] =

d2 2αd 2 (u0 )31 − f1 , ε ε

and M50 [1] = M12 [1]−1 , we obtain u1 = M50 [1]u2 + z[1]. where z[1] = M50 [1] ·Y0 [1]. Now for n = 2 we get 2

3αd (u0 )22 u2 u3 − 2u2 + u1 + M˜2 u2 − ε 2α β d2 d2 + (u0 )32 d 2 + u2 − f2 = 0, ε ε ε

(1.8)

that is, 2

3αd u3 − 2u2 + M50 [1]u2 + z[1] + M˜2 u2 − (u0 )22 u2 ε 2α β d2 d2 + (u0 )32 d 2 + u2 − f2 = 0, (1.9) ε ε ε so that denoting 2

2

βd αd (u0 )22 Id − Id , M12 [2] = 2Id − M˜ 2 − M50 [1] + 3 ε ε Y0 [2] =

d2 2αd 2 (u0 )32 − f2 , ε ε

and M50 [2] = M12 [2]−1 , we obtain u2 = M50 [2]u3 + z[2], where z[2] = M50 [2] · (Y0 [2] + z[1]). Proceeding in this fashion, for the line n we obtain 2

3αd (u0 )2n un un+1 − 2un + M50 [n − 1]un + z[n − 1] + M˜2 un − ε 2α β d2 d2 + (u0 )3n d 2 + un − fn = 0, (1.10) ε ε ε

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

■

9

so that denoting 2

2

βd αd (u0 )2n Id − Id , M12 [n] = 2Id − M˜ 2 − M50 [n − 1] + 3 ε ε and also denoting Y0 [n] =

2αd 2 d2 (u0 )3n − fn , ε ε

and M50 [n] = M12 [n]−1 , we obtain un = M50 [n]un+1 + z[n], where z[n] = M50 [n] · (Y0 [n] + z[n − 1]). Observe that we have uN = θ , where θ denotes the zero matrix (N1 − 1) × 1, so that we may calculate uN−1 = M50 [N − 1] · uN + z[N − 1], and, uN−2 = M50 [N − 2] · uN−1 + z[N − 2], and so on, up to obtaining u1 = M50 [1] · u2 + z[1]. The next step is to replace {(u0 )n } by {un } and thus to repeat the process until convergence is achieved. This is the Newton’s Method, what seems to be relevant is the way we inverted the big matrix ((N1 − 1) · (N − 1)) × ((N1 − 1) · (N − 1)), in fact instead of inverting it directly we have inverted N − 1 matrices (N1 − 1) × (N1 − 1) through an application of the generalized method of lines.

1.3.3

Numerical results for the concerning partial differential equation

We solve the equation ε∇2 u − αu3 + β u + 1 = 0,

u = 0,

in Ω = [0, 1] × [0, 1] (1.11) on ∂ Ω,

through the algorithm specified in the last section. We consider α = β = 1. For ε = 1.0 see Figure 1.5, for ε = 0.0001 see Figure 1.6.

10

■ The Method of Lines and Duality Principles for Non-Convex Models

0.08

0.06

0.04

0.02

0 1 1 0.8

0.5

0.6 0.4 0.2 0

0

Figure 1.5: The solution u(x, y) for ε = 1.0.

1.6

1.4

1.2

1

0.8 1 1 0.8

0.5

0.6 0.4 0.2 0

0

Figure 1.6: The solution u(x, y) for ε = 0.0001.

1.4

A numerical example concerning a Ginzburg-Landau type equation

We start by recalling that the generalized method of lines was originally introduced in the book entitled “Topics on Functional Analysis, Calculus of Variations and Duality” [22], published in 2011.

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

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11

Indeed, the present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [22, 17, 12, 13]. About the Sobolev spaces involved, we would mention [1, 2]. Concerning the applications, related models in physics are addressed in [4, 52]. We also emphasize that, in such a method, the domain of the partial differential equation in question is discretized in lines (or more generally, in curves), and the concerning solution is written on these lines as functions of boundary conditions and the domain boundary shape. In fact, in its previous format, this method consists of an application of a kind of partial finite differences procedure combined with the Banach fixed point theorem to obtain the relation between two adjacent lines (or curves). In the present article, we propose an improvement concerning the way we truncate the series solution obtained through an application of the Banach fixed point theorem to find the relation between two adjacent lines. The results obtained are very good even as a typical parameter ε > 0 is very small. In the next lines and sections, we develop in detail such a numerical procedure.

1.4.1

About the concerning improvement for the generalized method of lines

Let Ω ⊂ R2 where Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}. Consider the problem of solving the partial differential equation 2 2 −ε ∂∂ r2u + 1r ∂∂ ur + r12 ∂∂ θu2 + αu3 − β u = f , in Ω, u = u0 (θ ), on ∂ Ω1 , u = u f (θ ), on ∂ Ω2 .

(1.12)

Here Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ε > 0, α > 0, β > 0, and f ≡ 1, on Ω. In a partial finite differences scheme, such a system stands for un+1 − 2un + un−1 1 un − un−1 1 ∂ 2 un + + + αu3n − β un = fn , −ε d2 tn d tn2 ∂ θ 2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0,

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The Method of Lines and Duality Principles for Non-Convex Models

and uN = 0. Here N is the number of lines and d = 1/N. In particular, for n = 1 we have u2 − 2u1 + u0 1 (u1 − u0 ) 1 ∂ 2 u1 + + 2 + αu31 − β u1 = f1 , −ε d2 t1 d t1 ∂ θ 2 so that d2 1 1 ∂ 2 u1 2 3 u1 = u2 + u1 + u0 + (u1 − u0 ) d + 2 d + (−αu1 + β u1 − f1 ) /3.0, t1 ε t1 ∂ θ 2 We solve this last equation through the Banach fixed point theorem, obtaining u1 as a function of u2 . Indeed, we may set u01 = u2 and uk+1 1

1 ∂ 2 uk1 2 1 d = u2 + uk1 + u0 + (uk1 − u0 ) d + 2 t1 t1 ∂ θ 2 d2 k 3 k +(−α(u1 ) + β u1 − f1 ) /3.0, ε

(1.13)

∀k ∈ N. Thus, we may obtain u1 = lim uk1 ≡ H1 (u2 , u0 ). k→∞

Similarly, for n = 2, we have 1 ∂ 2 u2 2 1 d u2 = u3 + u2 + H1 (u2 , u0 ) + (u2 − H1 (u2 , u0 )) d + 2 t1 t1 ∂ θ 2 d2 3 /3.0, (1.14) +(−αu2 + β u2 − f2 ) ε We solve this last equation through the Banach fixed point theorem, obtaining u2 as a function of u3 and u0 . Indeed, we may set u02 = u3

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

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13

and = uk+1 2

1 1 ∂ 2 uk2 2 u3 + uk2 + H1 (uk2 , u0 ) + (uk2 − H1 (uk2 , u0 )) d + 2 d t2 t2 ∂ θ 2 d2 k 3 k +(−α(u2 ) + β u2 − f2 ) /3.0, (1.15) ε

∀k ∈ N. Thus, we may obtain u2 = lim uk2 ≡ H2 (u3 , u0 ). k→∞

Now reasoning inductively, having un−1 = Hn−1 (un , u0 ), we may get 1 1 ∂ 2 un 2 d un = un+1 + un + Hn−1 (un , u0 ) + (un − Hn−1 (un , u0 )) d + 2 tn tn ∂ θ 2 d2 +(−αu3n + β un − fn ) /3.0, (1.16) ε We solve this last equation through the Banach fixed point theorem, obtaining un as a function of un+1 and u0 . Indeed, we may set u0n = un+1 and uk+1 = n

1 1 ∂ 2 ukn 2 d un+1 + ukn + Hn−1 (ukn , u0 ) + (ukn − Hn−1 (ukn , u0 )) d + 2 tn tn ∂ θ 2 d2 k 3 k /3.0, (1.17) +(−α(un ) + β un − fn ) ε

∀k ∈ N. Thus, we may obtain un = lim ukn ≡ Hn (un+1 , u0 ). k→∞

We have obtained un = Hn (un+1 , u0 ), ∀n ∈ {1, · · · , N − 1}. In particular, uN = u f (θ ), so that we may obtain uN−1 = HN−1 (uN , u0 ) = HN−1 (0) ≡ FN−1 (uN , u0 ) = FN−1 (u f (θ ), u0 (θ )).

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The Method of Lines and Duality Principles for Non-Convex Models

Similarly, uN−2 = HN−2 (uN−1 , u0 ) = HN−2 (HN−1 (uN , u0 )) = FN−2 (uN , u0 ) = FN−1 (u f (θ ), u0 (θ )),

an so on, up to obtaining u1 = H1 (u2 ) ≡ F1 (uN , u0 ) = F1 (u f (θ ), u0 (θ )). The problem is then approximately solved.

1.4.2

Software in Mathematica for solving such an equation

We recall that the equation to be solved is a Gingurg-Landau type one, where 2 2 −ε ∂∂ r2u + 1r ∂∂ ur + r12 ∂∂ θu2 + αu3 − β u = f , in Ω, (1.18) u = 0, on ∂ Ω1 , u = u f (θ ), on ∂ Ω2 . Here Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ε > 0, α > 0, β > 0, and f ≡ 1, on Ω. In a partial finite differences scheme, such a system stands for un+1 − 2un + un−1 1 un − un−1 1 ∂ 2 un + + 2 + αu3n − β un = fn , −ε d2 tn d tn ∂ θ 2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0, and uN = u f [x]. Here N is the number of lines and d = 1/N. At this point we present the concerning software for an approximate solution. Such a software is for N = 10 (10 lines) and u0 [x] = 0.

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

************************************* 1. m8 = 10; (N = 10 lines) 2. d = 1/m8; 3. e1 = 0.1; (ε = 0.1) 4. A = 1.0; 5. B = 1.0; 6. For[i = 1, i < m8, i + +, f [i] = 1.0]; ( f ≡ 1, on Ω) 7. a = 0.0; 8. For[i = 1, i < m8, i + +, Clear[b, u]; t[i] = 1 + i ∗ d; b[x− ] = u[i + 1][x]; 9. For[k = 1, k < 30, k + +, (we have fixed the number of iterations) 1 (b[x] − a) ∗ d z = u[i + 1][x] + b[x] + a + t[i] 2 + t[i]1 2 D[b[x], {x, 2}] ∗ d 2 + (−A ∗ b[x]3 + B ∗ u[x] + f [i]) ∗ de1 /3.0; z= Series[z, {u[i + 1][x], 0, 3}, {u[i + 1]′ [x], 0, 1}, {u[i + 1]′′ [x], 0, 1}, {u[i + 1]′′′ [x], 0, 0}, {u[i + 1]′′′′ [x], 0, 0}]; z = Normal[z], z = Expand[z]; b[x− ] = z]; 10. a1 [i] = z; 11. Clear[b]; 12. u[i + 1][x− ] = b[x]; 13. a = a1 [i] ]; 14. b[x− ] = u f [x]; 15. For[i = 1, i < m8, i + +, A1 = a1 [m8 − i]; A1 = Series[A1 , {u f [x], 0, 3}, {u′f [x], 0, 1}, {u′′f [x], 0, 1}, {u′′′f [x], 0, 0}, {u′′′′ f [x], 0, 0}];

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15

16

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The Method of Lines and Duality Principles for Non-Convex Models

A1 = Normal[A1 ]; A1 = Expand[A1 ]; u[m8 − i][x− ] = A1 ; b[x− ] = A1 ]; Print[u[m8/2][x]]; ************************************* The numerical expressions for the solutions of the concerning N lines are given by u[1][x] = 0.47352 + 0.00691u f [x] − 0.00459u f [x]2 + 0.00265u f [x]3 +0.00039(u′′f )[x] − 0.00058u f [x](u′′f )[x] + 0.00050u f [x]2 (u′′f )[x] −0.000181213u f [x]3 (u′′f )[x]

(1.19)

u[2][x] = 0.76763 + 0.01301u f [x] − 0.00863u f [x]2 + 0.00497u f [x]3 +0.00068(u′′f )[x] − 0.00103u f [x](u′′f )[x] + 0.00088u f [x]2 (u′′f )[x] −0.00034u f [x]3 (u′′f )[x]

(1.20)

u[3][x] = 0.91329 + 0.02034u f [x] − 0.01342u f [x]2 + 0.00768u f [x]3 +0.00095(u′′f )[x] − 0.00144u f [x](u′′f )[x] + 0.00122u f [x]2 (u′′f )[x] −0.00051u f [x]3 (u′′f )[x]

(1.21)

u[4][x] = 0.97125 + 0.03623u f [x] − 0.02328u f [x]2 + 0.01289u f [x]3 +0.00147331(u′′f )[x] −0.00223u f [x](u′′f )[x] + 0.00182u f [x]2 (u′′f )[x] −0.00074u f [x]3 (u′′f )[x]

(1.22)

u[5][x] = 1.01736 + 0.09242u f [x] − 0.05110u f [x]2 + 0.02387u f [x]3 +0.00211(u′′f )[x] − 0.00378u f [x](u′′f )[x] + 0.00292u f [x]2 (u′′f )[x] −0.00132u f [x]3 (u′′f )[x]

(1.23)

u[6][x] = 1.02549 + 0.21039u f [x] − 0.09374u f [x]2 + 0.03422u f [x]3 +0.00147(u′′f )[x] − 0.00634u f [x](u′′f )[x] + 0.00467u f [x]2 (u′′f )[x] −0.00200u f [x]3 (u′′f )[x]

(1.24)

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

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17

u[7][x] = 0.93854 + 0.36459u f [x] − 0.14232u f [x]2 + 0.04058u f [x]3 +0.00259(u′′f )[x] − 0.00747373u f [x](u′′f )[x] + 0.0047969u f [x]2 (u′′f )[x] −0.00194u f [x]3 (u′′f )[x]

(1.25)

u[8][x] = 0.74649 + 0.57201u f [x] − 0.17293u f [x]2 + 0.02791u f [x]3 +0.00353(u′′f )[x] − 0.00658u f [x](u′′f )[x] + 0.00407u f [x]2 (u′′f )[x] −0.00172u f [x]3 (u′′f )[x]

(1.26)

u[9][x] = 0.43257 + 0.81004u f [x] − 0.13080u f [x]2 + 0.00042u f [x]3 +0.00294(u′′f )[x] − 0.00398u f [x](u′′f )[x] + 0.00222u f [x]2 (u′′f )[x] −0.00066u f [x]3 (u′′f )[x]

1.5

(1.27)

The generalized method of lines for a more general domain

This section develops the generalized method of lines for a more general type of domain. In our previous publications [22, 17], we highlighted that the method addressed there may present a relevant error as a parameter ε > 0 is too small, that is, as ε is about 0.01, 0.001, or even smaller. In the present section, we develop a solution for such a problem through a proximal formulation suitable for a large class of non-linear elliptic PDEs. At this point, we reintroduce the generalized method of lines, originally presented in F. Botelho [22]. In the present context, we add new theoretical and applied results to the original presentation. Especially the computations are all completely new. Consider first the equation ε∇2 u + g(u) + f = 0, in Ω ⊂ R2 ,

(1.28)

with the boundary conditions u = 0 on Γ0 and u = u f , on Γ1 . From now on we assume that u f , g and f are smooth functions (we mean C∞ functions), unless otherwise specified. Here Γ0 denotes the internal boundary of Ω and Γ1 the external one. Consider the simpler case where Γ1 = 2Γ0 ,

18

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The Method of Lines and Duality Principles for Non-Convex Models

and suppose there exists r(θ ), a smooth function such that Γ0 = {(θ , r(θ )) | 0 ≤ θ ≤ 2π}, being r(0) = r(2π). In polar coordinates the above equation may be written as ∂ 2u 1 ∂ u 1 ∂ 2u + + g(u) + f = 0, in Ω, + ∂ r2 r ∂ r r2 ∂ θ 2

(1.29)

and u = 0 on Γ0 and u = u f , on Γ1 . Define the variable t by t=

r . r(θ )

Also defining u¯ by u(r, θ ) = u(t, ¯ θ ), dropping the bar in u, ¯ equation (1.28) is equivalent to ∂ 2u ∂t 2

1 ∂u f2 (θ ) t ∂t 1 ∂ 2u f4 (θ ) ∂ 2 u + f3 (θ ) + 2 t ∂ θ ∂t t ∂θ2 + f5 (θ )(g(u) + f ) = 0, +

in Ω. Here f2 (θ ), f3 (θ ), f4 (θ ) and f5 (θ ) are known functions. More specifically, denoting f1 (θ ) = we have: f2 (θ ) = 1 +

−r′ (θ ) , r(θ ) f1′ (θ ) , 1 + f1 (θ )2

f3 (θ ) =

2 f1 (θ ) , 1 + f1 (θ )2

f4 (θ ) =

1 . 1 + f1 (θ )2

and

(1.30)

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

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19

Observe that t ∈ [1, 2] in Ω. Discretizing in t (N equal pieces which will generate N lines) we obtain the equation un+1 − 2un + un−1 (un − un−1 ) 1 + f2 (θ ) d2 d tn ∂ 2 un f4 (θ ) ∂ (un − un−1 ) 1 f3 (θ ) + + ∂θ tn d ∂ θ 2 tn2 1 1 + f5 (θ ) g(un ) + fn = 0, ε ε

(1.31)

∀n ∈ {1, ..., N − 1}. Here, un (θ ) corresponds to the solution on the line n. At this point we introduce a proximal approach, initially around the point{(U0 )n } through a constant K > 0 to be specified. In partial finite differences, such a system stands for un+1 − 2un + un−1 (un − un−1 ) 1 + f2 (θ ) d2 d tn ∂ (un − un−1 ) 1 ∂ 2 un f4 (θ ) + f3 (θ ) + ∂θ tn d ∂ θ 2 tn2 1 1 + f5 (θ ) g(un ) + fn ε ε K K − un + (U0 )n = 0, ε ε

(1.32)

∀n ∈ {1, ..., N − 1}. Here, un (θ ) corresponds to the solution on the line n. Thus denoting ε = e1 we may obtain un+1 − 2un + un−1 −

K un + Tn + fˆn = 0, e1

where Tn =

and

(un − un−1 ) 1 f2 (θ )d 2 d tn ∂ (un − un−1 ) 1 ∂ 2 un f4 (θ ) 2 + d f3 (θ )d 2 + ∂θ tn d ∂ θ 2 tn2 d2 + f5 (θ )(g(un ) e1 2

d fˆn = ( f5 (θ ) fn + K(U0 )n ) , e1 ∀n ∈ {1, · · · , N − 1}.

(1.33)

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The Method of Lines and Duality Principles for Non-Convex Models

In particular, for n = 1, we have u2 − 2u1 + u0 − K u1

d2 + T1 + fˆ1 . e1

Hence u1 = a1 u2 + b1 u0 + c1 T1 + F1 + Er1 , where

−1 d2 a1 = 2 + K , e1 b1 = a1 c1 = a1 F1 = a1 fˆ1 . Er1 = 0.

With such a result for n = 2 we have u3 − 2u2 + u1 − K u2

d2 + T2 + fˆ2 . e1

Hence u2 = a2 u3 + b2 u0 + c2 T2 + F2 + Er2 , where

−1 d2 , a2 = 2 + K − a1 e1 b2 = a2 b1 c1 = a2 (c1 + 1) F1 = a2 ( fˆ2 + F1 ) Er2 = a2 (c1 (T1 − T2 )).

Reasoning inductively, having un−1 = an−1 un + bn−1 u0 + cn−1 Tn−1 + Fn−1 + Ern−1 , for the line n, we have un+1 − 2un + un−1 − K un−1

d2 + Tn + fˆn . e1

Hence un = an un+1 + bn u0 + cn Tn + Fn + Ern ,

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

where

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21

−1 d2 an = 2 + K − an−1 , e1 bn = an bn−1 cn = an (cn−1 + 1) Fn = an ( fˆn + Fn−1 ) Ern = an (Ern−1 + cn−1 (Tn−1 − Tn )),

∀n ∈ {1, · · · , N − 1}. In particular we have uN = u f . Thus, for n = N − 1 we have got uN−1 ≈ aN−1 u f + bN−1 u0 + cN−1 TN−1 (uN−1 , u f ) + FN−1 . This last equation is an ODE in uN−1 which may be easily solved with the boundary conditions uN−1 (0) = uN−1 (2π) and u′N−1 (0) = u′N−1 (2π). Having uN−1 we may obtain uN−2 through n = N − 2. Observe that, uN−2 ≈ aN−2 uN−1 + bN−2 u0 + cN−2 TN−2 (uN−2 , uN−1 ) + FN−2 . This last equation is again an ODE in uN−2 which may be easily solved with the boundary conditions uN−2 (0) = uN−2 (2π) and u′N−2 (0) = u′N−2 (2π). We may continue inductively with such a reasoning up to obtaining u1 . The next step is to replace {(U0 )n } by {un } and repeat the process until an appropriate convergence criterion is satisfied. The problem is then solved. At this point we present a software in MATHEMATICA based on this last algorithm. Some changes has been implemented concerning this last previous conception, in particular in order to make it suitable as a software in MATHEMATICA.

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The Method of Lines and Duality Principles for Non-Convex Models

Here the concerning software. ****************************************** 1. m8 = 10; d = 1.0/m8; K = 5.0; e1 = 1.0; A = 1.0; B = 1.0; Clear[t3]; 2. For[i = 1, i < m8 + 1, i + +, uo[i] = 0.0]; 3. For[k = 1, k < 20, k + +, Print[k]; a[1] = 1/(2.0 + K ∗ d 2 /e1); b[1] = a[1]; c[1] = a[1] ∗ (K ∗ uo[1] + 1.0 ∗ t3 ∗ f 5[x]) ∗ d 2 /e1; 4. For[i = 2, i < m8, i + +, a[i] = 1/(2.0 + K ∗ d 2 /e1 − a[i − 1]); b[i] = a[i] ∗ (b[i − 1] + 1); c[i] = a[i] ∗ (c[i − 1] + (K ∗ uo[i] + 1.0 ∗ t3 ∗ f 5[x]) ∗ d 2 /e1)]; u[m8] = u f [x]; 5. For[i = 1, i < m8, i + +, Print[i]; t[m8 − i] = 1 + (m8 − i) ∗ d; A1 = a[m8 − i] ∗ u[m8 − i + 1]+ b[m8 − i] ∗ (−A ∗ u[m8 − i + 1]3 + B ∗ u[m8 − i + 1]) ∗ d 2 /e1) ∗ f 5[x] ∗ t3 + c[m8 − i] +d 2 ∗ b[m8 − i] ∗ ( f 4[x] ∗ D[u[m8 − i + 1], x, 2]/t[m8 − i]2 ) ∗ t3 + f 3[x] ∗t3 ∗ b[m8 − i]/t[m8 − i] ∗ D[(uo[m8 − i + 1] − uo[m8 − i])/d, x] ∗ d 2 + f 2[x] ∗ t3 ∗ 1/t[m8 − i] ∗ b[m8 − i] ∗ d 2 (uo[m8 − i + 1] − uo[m8 − i])/d; A1 = Expand[A1]; A1 = Series[A1, {t3, 0, 2}, {u f [x], 0, 3}, {u f ′ [x], 0, 1},

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

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23

{u f ′′ [x], 0, 1}, {u f ′′′ [x], 0, 0}, {u f ′′′′ [x], 0, 0}, { f 5[x], 0, 2}, { f 4[x], 0, 2}, { f 3[x], 0, 2}, { f 2[x], 0, 2}, { f 5′ [x], 0, 0}, { f 4′ [x], 0, 0}, { f 3′ [x], 0, 0}, { f 2′ [x], 0, 0}, { f 5′′ [x], 0, 0}, { f 4′′ [x], 0, 0}, { f 3′′ [x], 0, 0}, { f 2′′ [x], 0, 0}]; A1 = Normal[A1]; u[m8 − i] = Expand[A1]]; 6. For[i = 1, i < m8 + 1, i + +, uo[i] = u[i]]; Print[Expand[u[m8/2]]]; ] t3 = 1; For[i = 1, i ¡ m8, i++, Print[“u[”, i, “]=”, u[i]]] *************************************************** Here we present the lines expressions for the lines n = 2, n = 4, n = 6 and n = 8 of a total of N = 10 lines, u[2]

=

0.08 f 5[x] − 0.00251535 f 2[x] f 5[x] + 0.00732449 f 5[x]2 + 0.2u f [x] +0.0534377 f 2[x]u f [x] − 0.00202221 f 2[x]2 u f [x] + 0.0503925 f 5[x]u f [x] +0.0035934 f 2[x] f 5[x]u f [x] + 0.00580877 f 5[x]2 u f [x] −0.00741623 f 5[x]2 u f [x]2 − 0.0276371 f 5[x]u f [x]3 −0.00455457 f 2[x] f 5[x]u f [x]3 − 0.0103934 f 5[x]2 u f [x]3 +0.0534377 f 3[x](u f ′ )[x] − 0.00404441 f 2[x] f 3[x](u f ′ )[x] +0.0035934 f 3[x] f 5[x](u f ′ )[x] − 0.00506561 f 3[x] f 5[x]u f [x]2 (u f ′ )[x] −0.00202221 f 3[x]2 (u f ′′ )[x] + 0.0206072 f 4[x](u f ′′ )[x] +0.00161163 f 2[x] f 4[x](u f ′′ )[x] + 0.00479841 f 4[x] f 5[x](u f ′′ )[x] −0.00784991 f 4[x] f 5[x]u f [x]2 (u f ′′ )[x]

(1.34)

24

■

u[4]

The Method of Lines and Duality Principles for Non-Convex Models

=

0.12 f 5[x] − 0.0115761 f 2[x] f 5[x] + 0.0101622 f 5[x]2 + 0.4u f [x] +0.0747751 f 2[x]u f [x] − 0.00755983 f 2[x]2 u f [x] + 0.0868571 f 5[x]u f [x] −0.00156623 f 2[x] f 5[x]u f [x] + 0.00847093 f 5[x]2 u f [x] −0.013132 f 5[x]2 u f [x]2 − 0.0534765 f 5[x]u f [x]3 −0.00387899 f 2[x] f 5[x]u f [x]3 − 0.0177016 f 5[x]2 u f [x]3 +0.0747751 f 3[x](u f ′ )[x] − 0.0151197 f 2[x] f 3[x](u f ′ )[x] −0.00156623 f 3[x] f 5[x](u f ′ )[x] + 0.00341553 f 3[x] f 5[x]u f [x]2 (u f ′ )[x] −0.00755983 f 3[x]2 (u f ′′ )[x] + 0.0323379 f 4[x](u f ′′ )[x] +0.000516904 f 2[x] f 4[x](u f ′′ )[x] + 0.0063856 f 4[x] f 5[x](u f ′′ )[x] −0.0116439 f 4[x] f 5[x]u f [x]2 (u f ′′ )[x]

u[6]

=

(1.35)

0.12 f 5[x] − 0.0178517 f 2[x] f 5[x] + 0.00790756 f 5[x]2 + 0.6u f [x] +0.0704471 f 2[x]u f [x] − 0.010741 f 2[x]2 u f [x] + 0.0978209 f 5[x]u f [x] −0.00934984 f 2[x] f 5[x]u f [x] + 0.00692381 f 5[x]2 u f [x] −0.0131832 f 5[x]2 u f [x]2 − 0.0703554 f 5[x]u f [x]3 +0.00171175 f 2[x] f 5[x]u f [x]3 − 0.0174038 f 5[x]2 u f [x]3 +0.0704471 f 3[x](u f ′ )[x] − 0.0214821 f 2[x] f 3[x](u f ′ )[x] −0.00934984 f 3[x] f 5[x](u f ′ )[x] + 0.0199603 f 3[x] f 5[x]u f [x]2 (u f ′ )[x] −0.010741 f 3[x]2 (u f ′′ )[x] + 0.0329474 f 4[x](u f ′′ )[x] −0.00316522 f 2[x] f 4[x](u f ′′ )[x] + 0.00478074 f 4[x] f 5[x](u f ′′ )[x] −0.00994569 f 4[x] f 5[x]u f [x]2 (u f ′′ )[x]

u[8]

=

(1.36)

0.08 f 5[x] − 0.015345 f 2[x] f 5[x] + 0.002891 f 5[x]2 + 0.8u f [x] +0.0445896 f 2[x]u f [x] − 0.00871597 f 2[x]2 u f [x] + 0.0723048 f 5[x]u f [x] −0.0117866 f 2[x] f 5[x]u f [x] + 0.00260131 f 5[x]2 u f [x] −0.00564143 f 5[x]2 u f [x]2 − 0.0619403 f 5[x]u f [x]3 +0.00783579 f 2[x] f 5[x]u f [x]3 − 0.00745182 f 5[x]2 u f [x]3 +0.0445896 f 3[x](u f ′ )[x] − 0.0174319 f 2[x] f 3[x](u f ′ )[x] −0.0117866 f 3[x] f 5[x](u f ′ )[x] + 0.0297437 f 3[x] f 5[x]u f [x]2 (u f ′ )[x] −0.00871597 f 3[x]2 (u f ′′ )[x] + 0.0220765 f 4[x](u f ′′ )[x] −0.00363515 f 2[x] f 4[x](u f ′′ )[x] + 0.00171483 f 4[x] f 5[x](u f ′′ )[x] −0.00389297 f 4[x] f 5[x]u f [x]2 (u f ′′ )[x].

(1.37)

The Generalized Method of Lines Applied to a Ginzburg-Landau Type Equation

1.6

■

25

Conclusion

In this article, we present an advance concerning the computation of a solution for a partial differential equation through the generalized method of lines. In particular, in its previous versions, we used to truncate the series in d 2 however, we have realized the results are much better by taking line solutions in series for u f [x] and its derivatives, as it is indicated in the present software. This is a little different from the previous procedure, but with a great result in improvement as the parameter ε > 0 is small. Indeed, with a sufficiently large N (number of lines), we may obtain very good results even as ε > 0 is very small.

Chapter 2

An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines

2.1

Introduction

This chapter develops an approximate proximal approach for the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves), and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. Considering such a context, along the text we develop an approximate numerical procedure of proximal nature applicable to a large class of models in physics and engineering. Finally, in the last sections, we present numerical examples and results related to a Ginzburg-Landau type equation. Remark 2.1.1 This chapter has been published in a similar article format by the MDPI Journal Mathematics, reference: Fabio Silva Botelho, Mathematics 2022, 10(16), 2950; https://doi.org/10.3390/math10162950 - 16 Aug 2022.

An Approximate Proximal Procedure through the Method of Lines

■

27

We start by recalling that the generalized method of lines was originally introduced in the book entitled “Topics on Functional Analysis, Calculus of Variations and Duality” [22], published in 2011. Indeed, the present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [22, 17, 12, 13]. About the Sobolev spaces involved, we would mention [1, 2]. Concerning the applications, related models in physics are addressed in [4, 52]. We also emphasize that, in such a method, the domain of the partial differential equation in question is discretized in lines (or more generally, in curves), and the concerning solution is written on these lines as functions of boundary conditions and the domain boundary shape. In fact, in its previous format, this method consists of an application of a kind of partial finite differences procedure combined with the Banach fixed point theorem to obtain the relation between two adjacent lines (or curves). In the present article, we propose an approximate approach and a related iterative procedure of proximal nature. We highlight this as a proximal method inspired by some models concerning duality principles in D.C optimization in the calculus of variations, such as those found in Toland [81]. In the following lines and sections, we develop in detail such a numerical procedure. With such statements in mind, let Ω ⊂ R2 be an open, bounded and connected set where Ω = {(x, y) ∈ R2 : y1 (x) ≤ y ≤ y2 (x), a ≤ x ≤ b}. Here, we assume, y1 , y2 : [a, b] → R are continuous functions. Consider the Ginzburg-Landau type equation, defined by −ε∇2 u + αu3 − β u = f , in Ω, u = 0, on ∂ Ω,

(2.1)

where ε > 0, α > 0, β > 0 and f ∈ L2 (Ω). Also, u ∈ W01,2 (Ω) and the equation in question must be considered in a distributional sense. In the next section, we address the problem of approximately solving this concerning equation. We highlight the methods and the ideas exposed are applicable to a large class of similar models in physics and engineering.

2.2

The numerical method

We discretize the interval [a, b] into N same measure sub-intervals, through a partition P = {x0 = a, x1 , · · · , xN = b},

28

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The Method of Lines and Duality Principles for Non-Convex Models

where xn = a + nd, ∀n ∈ {1, · · · , N − 1}. Here d=

(b − a) . N

Through such a procedure, we generate N vertical lines parallel to the Cartesian axis 0y, so that for each line n based on the point xn we are going to compute an approximate solution un (y) corresponding to values of u on such a line. Considering this procedure, the equation system obtained in partial finite differences (please see [71], for concerning models in finite differences) is given by un+1 − 2un + un−1 ∂ 2 un −ε + + αu3n − β un = fn , d2 ∂ y2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0, and uN = 0. Let K > 0 be an appropriate constant to be specified. In a proximal approach, considering an initial solution {(u0 )n } we redefine the system of equations in question as below indicated. un+1 − 2un + un−1 ∂ 2 un + + αu3n − β un + Kun − K(u0 )n = fn , −ε d2 ∂ y2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0, and uN = 0. Hence, we may denote d2 d2 un + un−1 + T (un ) + f˜n = 0, un+1 − 2 + K ε ε where T (un ) = −αu3n + β un and f˜n = K(u0 )n + fn , ∀n ∈ {1, · · · , N − 1}.

d 2 ∂ 2 un 2 + d , ε ∂ y2

An Approximate Proximal Procedure through the Method of Lines

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29

In particular, for n = 1, we get d2 d2 u2 − 2 + K u1 + T (u1 ) + f˜1 = 0, ε ε so that u1 = a1 u2 + b1 T (u2 ) + c1 + E1 , where a1 =

1 2

2 + K dε

,

b1 = a1 , and

d2 ε and the error E1 , proportional to 1/K, is given by c1 = a1 f˜1

E1 = b1 (T (u1 ) − T (u2 )). Now, reasoning inductively, having un−1 = an−1 un + bn−1 T (un ) + cn−1 + En−1 for the line n, we have d2 un + an−1 un + bn−1 T (un ) + cn−1 + En−1 un+1 − 2 + K ε 2

d +T (un ) + f˜n = 0, ε

(2.2)

so that un = an un+1 + bn T (un+1 ) + cn + En , where an =

1 2 2 + K dε

− an−1

,

bn = an (bn−1 + 1), and

d2 cn = an cn−1 + f˜n ε

and the error En , is given by En = an (En−1 + bn (T (un ) − T (un+1 ))), ∀n ∈ {1, · · · , N − 1}.

30

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The Method of Lines and Duality Principles for Non-Convex Models

In particular, for n = N − 1, we have uN = 0 so that, uN−1 ≈ aN−1 uN + bN−1 T (uN ) + cN−1 d2 ∂ 2 uN−1 2 3 d + b (−αu + β u ) + cN−1 ≈ aN−1 uN + bN−1 N N−1 N ∂ y2 ε ∂ 2 uN−1 2 = bN−1 d + cN−1 . (2.3) ∂ y2 This last equation is an ODE from which we may easily obtain uN−1 with the boundary conditions uN−1 (y1 (xN−1 )) = uN−1 (y2 (xN−1 )) = 0. Having uN−1 , we may obtain uN−2 though the equation uN−2

≈ aN−2 uN−1 + bN−2 T (uN−1 ) + cN−2 ≈ aN−2 uN−1 + bN−2

d2 ∂ 2 uN−2 2 d + bN−2 (−αu3N−1 + β uN−1 ) + cN−2 , (2.4) 2 ∂y ε

with the boundary conditions uN−2 (y1 (xN−2 )) = uN−2 (y2 (xN−2 )) = 0. An so on, up to finding u1 . The next step is to replace {(u0 )n } by {un } and then to repeat the process until an appropriate convergence criterion is satisfied. The problem is then approximately solved.

2.3

A numerical example

We present numerical results for Ω = [0, 1] × [0, 1], α = β = 1, f ≡ 1 in Ω, N = 100, K = 50 and for ε = 0.1, 0.01 and 0.001. For such values of ε, please see Figures 2.1, 2.2 and 2.3, respectively. Remark 2.3.1 We observe that as ε > 0 decreases to the value 0.001 the solution approaches the constant value 1.3247 along the domain, up to the satisfaction of boundary conditions. This is expected, since this value is an approximate solution of equation u3 − u − 1 = 0.

An Approximate Proximal Procedure through the Method of Lines

■

31

Figure 2.1: Solution u for ε = 0.1.

Figure 2.2: Solution u for ε = 0.01.

2.4

A general proximal explicit approach

Based on the algorithm presented in the last section, we develop a software in MATHEMATICA to approximately solve the following equation. 2 2 −ε ∂∂ r2u + 1r ∂∂ ur + r12 ∂∂ θu2 + αu3 − β u = f , in Ω, (2.5) u = 0, on ∂ Ω1 , u = u f (θ ), on ∂ Γ2 .

32

■ The Method of Lines and Duality Principles for Non-Convex Models

Figure 2.3: Solution u for ε = 0.001.

Here Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, α = β = 1, K = 10, N = 100 and f ≡ 1, on Ω. At this point we present such a software in MATHEMATICA. ****************************************** 1. m8 = 100; 2. d = 1.0/m8; 3. K = 10.0; 4. e1 = 0.01; 5. A = 1.0; 6. B = 1.0; 7. For[i = 1, i < m8 + 1, i++, uo[i] = 0.0];

An Approximate Proximal Procedure through the Method of Lines

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33

8. For[k = 1, k < 150, k++, Print[k]; a[1] = 1/(2.0 + K ∗ d 2 /e1); b[1] = a[1]; c[1] = a[1]*(K*uo[1] + 1.0)*d 2 /e1; For[i = 2, i < m8, i++, a[i] = 1/(2.0 + K ∗ d 2 /e1 − a[i − 1]); b[i] = a[i]*(b[i - 1] + 1); c[i] = a[i]*(c[i - 1] + (K ∗ uo[i] + 1.0) ∗ d 2 /e1)]; 9. u[m8] = uf[x]; d1 = 1.0; 10. For[i = 1, i < m8, i++, t[m8 - i] = 1 + (m8 - i)*d; A1 = (a[m8 - i]*u[m8 - i + 1] + b[m8 - i]*(-A*u[m8 − i + 1]3 + B*u[m8 - i + 1])*d 2 /e1 ∗ d12 + c[m8 - i] + d 2 ∗ d12 *b[m8 - i]*(D[u[m8 - i + 1], x, 2]/t[m8 − i]2 ) + d12 ∗ 1/t[m8 − i]*b[m8 - i]* d 2 (uo[m8 - i + 1] - uo[m8 - i])/d)/(1.0); A1 = Expand[A1]; A1 = Series[ A1, {uf[x], 0, 3}, {uf’[x], 0, 1}, {uf”[x], 0, 1}, {uf”’[x], 0, 0}, {uf””[x], 0, 0}]; A1 = Normal[A1]; u[m8 - i] = Expand[A1]]; For[i = 1, i < m8 + 1, i++, uo[i] = u[i]]; d1 = 1.0; Print[Expand[u[m8/2]]]] ************************************ For such a general approach, for ε = 0.1, we have obtained the following lines (here x stands for θ ). u[10](x)

=

0.4780 + 0.0122 u f [x] − 0.0115 u f [x]2 + 0.0083 u f [x]3 + 0.00069u′′f [x] −0.0014 u f [x](u′′f )[x] + 0.0016 u f [x]2 u′′f [x] − 0.00092 u f [x]3 u′′f [x] (2.6)

34

■

The Method of Lines and Duality Principles for Non-Convex Models

0.7919 + 0.0241 u f [x] − 0.0225 u f [x]2 + 0.0163 u f [x]3 + 0.0012 u′′f [x]

u[20](x) =

−0.0025 u f [x](u′′f )[x] + 0.0030 u f [x]2 (u′′f )[x] − 0.0018 u f [x]3 (u′′f )[x]. (2.7) u[30](x) = 0.9823 + 0.0404 u f [x] − 0.0375 u f [x]2 + 0.0266 u f [x]3 + 0.00180 (u′′f )[x] −0.00362 u f [x](u′′f )[x] + 0.0043 u f [x]2 (u f ′′ )[x] − 0.0028 u f [x]3 (u f ′′ )[x] (2.8)

u[40](x)

=

1.0888 + 0.0698 u f [x] − 0.0632 u f [x]2 + 0.0433 u f [x]3 + 0.0026(u′′f )[x] −0.0051 u f [x](u′′f )[x] + 0.0061u f [x]2 (u′′f )[x] − 0.0043u f [x]3 (u′′f )[x] (2.9)

u[50](x) = 1.1316 + 0.1277 u f [x] − 0.1101 u f [x]2 + 0.0695 u f [x]3 + 0.0037 (u′′f )[x] −0.0073 u f [x](u′′f )[x] + 0.0084 u f [x]2 (u′′f )[x] − 0.0062 u f [x]3 (u′′f )[x]

(2.10)

u[60](x) = 1.1104 + 0.2389 u f [x] − 0.1866 u f [x]2 + 0.0988 u f [x]3 + 0.0053(u′′f )[x] −0.0099 u f [x](u′′f )[x] + 0.0105 u f [x]2 (u′′f )[x] − 0.0075 u f [x]3 (u′′f ])[x] (2.11) u[70](x) = 1.0050 + 0.4298 u f [x] − 0.273813 u f [x]2 + 0.0949 u f [x]3 + 0.0070 (u′′f )[x] −0.0116 u f [x](u′′f )[x] + 0.0102 u f [x]2 (u′′f )[x] − 0.0061 u f [x]3 (u′′f )[x] (2.12) u[80](x) = 0.7838 + 0.6855 u f [x] − 0.2892 u f [x]2 + 0.0161 u f [x]3 + 0.0075 (u′′f )[x] −0.0098u f [x](u′′f )[x] + 0.0063084u f [x]2 (u f ′′ )[x] − 0.0027u f [x]3 (u′′f )[x] (2.13)

u[90](x) = 0.4359 + 0.9077 u f [x] − 0.1621u f [x]2 − 0.0563 u f [x]3 + 0.0051 (u′′f )[x] −0.0047 u f [x](u′′f )[x] + 0.0023 u f [x]2 (u′′f )[x] − 0.00098 u f [x]3 (u′′f )[x] (2.14)

For ε = 0.01, we have obtained the following line expressions. u[10](x)

=

1.0057 + 2.07 ∗ 10−11 u f [x] − 1.85 ∗ 10−11 u f [x]2 + 1.13 ∗ 10−11 u f [x]3 +4.70 ∗ 10−13 (u′′f )[x] − 8.44 ∗ 10−13 u f [x](u′′f )[x] +7.85 ∗ 10−13 u f [x]2 (u′′f )[x] − 6.96 ∗ 10−14 u f [x]3 (u′′f )[x]

u[20](x)

=

(2.15)

1.2512 + 2.13 ∗ 10−10 u f [x] − 1.90 ∗ 10−10 u f [x]2 + 1.16 ∗ 10−10 u f [x]3 +3.94 ∗ 10−12 (u′′f )[x] − 7.09 ∗ 10−12 u f [x](u f ′′ )[x] + 6.61 ∗ 10−12 u f [x]2 (u′′f )[x] − 7.17 ∗ 10−13 u f [x]3 (u′′f )[x]

(2.16)

An Approximate Proximal Procedure through the Method of Lines

u[30](x)

■

35

1.3078 + 3.80 ∗ 10−9 u f [x] − 3.39 ∗ 10−9 u f [x]2 + 2.07 ∗ 10−9 u f [x]3

=

+5.65 ∗ 10−11 (u′′f )[x] − 1.018 ∗ 10−10 u f [x](u′′f )[x] +9.52 ∗ 10−11 u f [x]2 (u′′f )[x] − 1.27 ∗ 10−11 u f [x]3 (u′′f )[x]

u[40](x)

(2.17)

1.3208 + 7.82 ∗ 10−8 u f [x] − 6.98 ∗ 10−8 u f [x]2 + 4.27 ∗ 10−8 u f [x]3

=

+9.27 ∗ 10−10 (u′′f )[x] − 1.67 ∗ 10−9 u f [x](u′′f )[x] +1.57 ∗ 10− 9 u f [x]2 (u′′f )[x] − 2.62 ∗ 10−10 u f [x]3 (u′′f )[x]

u[50](x)

(2.18)

1.3238 + 1.67 ∗ 10−6 u f [x] − 1.49 ∗ 10−6 u f [x]2 + 9.15 ∗ 10−7 u f [x]3

=

+1.54 ∗ 10−8 (u′′f )[x] − 2.79 ∗ 10−8 u f [x](u′′f )[x] +2.64 ∗ 10−8 u f [x]2 (u′′f )[x] − 5.62 ∗ 10−9 u f [x]3 (u′′f )[x]

u[60](x)

=

(2.19)

1.32449 + 0.000036 u f [x] − 0.000032 u f [x]2 + 0.000019 u f [x]3 +2.51 ∗ 10−7 (u′′f )[x] − 4.57 ∗ 10−7 u f [x](u′′f )[x] +4.36 ∗ 10−7 u f [x]2 (u′′f )[x] − 1.21 ∗ 10−7 u f [x]3 (u′′f )[x]

u[70](x)

=

(2.20)

1.32425 + 0.00079 u f [x] − 0.00070 u f [x]2 + 0.00043 u f [x]3 +3.89 ∗ 10−6 (u′′f )[x] − 7.12 ∗ 10−6 u f [x](u′′f )[x] +6.89 ∗ 10−6 u f [x]2 (u′′f )[x] − 2.64 ∗ 10−6 u f [x]3 (u′′f )[x]

u[80](x)

=

(2.21)

1.31561 + 0.017 u f [x] − 0.015 u f [x]2 + 0.009 u f [x]3 +0.000053 (u′′f )[x] − 0.000098 u f [x](u′′f )[x] +0.000095 u f [x]2 (u′′f )[x] − 0.000051 u f [x]3 (u′′f )[x]

u[90](x)

=

(2.22)

1.14766 + 0.296u f [x] − 0.1991 u f [x]2 + 0.0638 u f [x]3 +0.00044 (u′′f )[x] − 0.00067 u f [x](u′′f )[x] +0.00046 u f [x]2 (u′′f )[x] − 0.00018 u f [x]3 (u′′f )[x]

(2.23)

Remark 2.4.1 We observe that as ε > 0 decreases to the value 0.01 the solution approaches the constant value 1.3247 along the domain, up to the satisfaction of boundary conditions. This is expected, since this value is an approximate solution of equation u3 − u − 1 = 0.

36

2.5

■

The Method of Lines and Duality Principles for Non-Convex Models

Conclusion

In this article, we develop an approximate numerical procedure related to the generalized method of lines. Such a procedure is proximal and involves a parameter K > 0 which minimizes the concerning numerical error of the initial approximation. We have presented numerical results concerning a Ginzburg-Landau type equation. The results obtained are consistent with those expected for such a mathematical model. In the last section, we present a software in MATHEMATICA for approximately solving a large class of similar systems of partial differential equations. Finally, for future research, we intend to extend the results for the NavierStokes system in fluid mechanics and related time dependent models in physics and engineering.

Chapter 3

Approximate Numerical Procedures for the Navier-Stokes System through the Generalized Method of Lines

3.1

Introduction

In this chapter, we develop approximate solutions for the time independent incompressible Navier-Stokes system through the generalized method of lines. We recall again, for such a method, the domain of the partial differential equation in question is discretized in lines, and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. We emphasize the first article part concerns the application and extension of an approximate proximal approach published in [18]. We develop an analogous algorithm as those presented in [18] but now for a Navier-Stokes system, which is more complex than the systems previously addressed. In this first step, we present an algorithm and respective software in MATLAB® . Furthermore, we have developed and presented related software in MATHEMATICA for a simpler type of domain but also concerning the mentioned proximal approach. Finally, in the last section, we present a software and related line expressions through the original conception of the generalized method of

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The Method of Lines and Duality Principles for Non-Convex Models

lines, so that in such related numerical examples, the main results are established through applications of the Banach fixed point theorem. Remark 3.1.1 We also highlight that the following two paragraphs in this article (a relatively small part) overlap with Chapter 28, starting on page 526, in the book by F.S. Botelho, [13], published in 2020, by CRC Taylor and Francis. However, we emphasize the present article includes substantial new parts, including a concerning software not included in the previous version of 2020. Another novelty in the present version is the establishment of appropriate boundary conditions for an elliptic system equivalent to original Navier-Stokes one. Such new boundary conditions and concerning results are indicated in Section 3.2. At this point we describe the system in question. Consider Ω ⊂ R2 an open, bounded and connected set with a regular (Lipschitzian) internal boundary denoted Γ0 , and a regular external one denoted by Γ1 . For a two-dimensional motion of a fluid on Ω, we denote by u : Ω → R the velocity field in the direction x of the Cartesian system (x, y), by v : Ω → R, the velocity field in the direction y and by p : Ω → R, the pressure one. Moreover, ρ denotes the fluid density, µ is the viscosity coefficient and g denotes the gravity field. Under such notation and statements, the time-independent incompressible Navier-Stokes system of partial differential equations stands for, µ∇2 u − ρ u ux − ρ v uy − px + ρ gx = 0, in Ω, (3.1) µ∇2 v − ρ u vx − ρ v vy − py + ρ gy = 0, in Ω, ux + vy = 0, in Ω, (

u = v = 0,

on Γ0 ,

u = u∞ , v = 0, p = p∞ ,

on Γ1

(3.2) At first we look for solutions (u, v, P) ∈ W 2,2 (Ω) × W 2,2 (Ω) × W 1,2 (Ω). We emphasize that details about such Sobolev spaces may be found in [1]. About the references, we emphasize that related existence, numerical and theoretical results for similar systems may be found in [39, 40, 41, 42] and [73], respectively. In particular [73] addresses extensively both theoretical and numerical methods and an interesting interplay between them. Moreover, related finite difference schemes are addressed in [71].

Approximate Numerical Procedures for the Navier-Stokes System

3.2

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39

Details about an equivalent elliptic system

Defining now P = p/ρ and ν = µ/ρ, consider again the Navier-Stokes system in the following format ν∇2 u − u∂x u − v∂y u − ∂x P + gx = 0, in Ω, (3.3) ν∇2 v − u∂x v − v∂y v − ∂y P + gy = 0, in Ω, ∂x u + ∂y v = 0, in Ω, (

u = v = 0,

on Γ0 ,

u = u∞ , v = 0, P = P∞ ,

on Γ1

(3.4) As previously mentioned, at first we look for solutions (u, v, P) ∈ W 2,2 (Ω) × W 2,2 (Ω) ×W 1,2 (Ω). We are going to obtain an equivalent Elliptic system with appropriate boundary conditions. Our main result is summarized by the following theorem. Theorem 3.2.1 Let Ω ⊂ R2 be an open, bounded, connected set with a regular (Lipschitzian) boundary. Assume u, v, P ∈ W 2,2 (Ω) are such that ν∇2 u − u ux − v uy − Px + gx = 0, ν∇2 v − u vx − v vy − Py + gy = 0, 2 ∇ P + u2x + v2y + 2uy vx − div g = 0, (

u = u0 , v = v0 ,

on ∂ Ω,

ux + vy = 0,

on ∂ Ω.

in Ω, in Ω,

(3.5)

in Ω,

(3.6) Suppose also the unique solution of equation in w ν∇2 w − u wx − v wy = 0, in Ω with the boundary conditions w = 0 on ∂ Ω, is w = 0, in Ω.

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Under such hypotheses, u, v, P solve the following Navier-Stokes system ν∇2 u − u ux − v uy − Px + gx = 0, in Ω, (3.7) ν∇2 v − u vx − v vy − Py + gy = 0, in Ω, ux + vy = 0, in Ω, (

u = u0 , v = v0 ,

on ∂ Ω,

ux + vy = 0,

on ∂ Ω.

(3.8)

Proof 3.1 In (3.5), taking the derivative in x of the first equation and adding with the derivative in y of the second equation, we obtain ν∇2 (ux + vy ) − u(ux + vy )x − v(ux + vy )y −∇2 P − u2x − v2y − 2uy vx + div g = 0, in Ω

(3.9)

From the hypotheses, u, v, P are such that ∇2 P + u2x + v2y + 2uy vx − div g = 0, in Ω, From this and (13.14), we get ν∇2 (ux + vy ) − u(ux + vy )x − v(ux + vy )y = 0, in Ω.

(3.10)

Denoting w = ux + vy , from this last equation we obtain ν∇2 w − uwx − vwy = 0, in Ω. From the hypothesis, the unique solution of this last equation with the boundary conditions w = 0, on ∂ Ω, is w = 0. From this and (3.10) we have ux + vy = 0, in Ω with the boundary conditions ux + vy = 0, on ∂ Ω. The proof is complete.

Remark 3.2.2 The process of obtaining such a system with a Laplace operator in P in the third equation is a standard and well known one. The novelty here is the identification of the corrected related boundary conditions obtained through an appropriate solution of equation (3.10).

Approximate Numerical Procedures for the Navier-Stokes System

3.3

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41

An approximate proximal approach

In this section, we develop an approximate proximal numerical procedure for the model in question. Such results are extensions of previous ones published in F.S. Botelho [18] now for the Navier-Stokes system context. More specifically, neglecting the gravity field, we solve the system of equations ν∇2 u − u∂x u − v∂y u − ∂x P = 0, in Ω, ν∇2 v − u∂x v − v∂y v − ∂y P = 0, in Ω, (3.11) 2 ∇ P + (∂x u)2 + (∂y v)2 + 2(∂y u)(∂x v) = 0, in Ω. We present a software similar to those presented in [18], with ν = 0.0177, and with Ω = [0, 1] × [0, 1] with the boundary conditions u = u0 = 0.65y(1 − y), v = v0 = 0, P = p0 = 0.15 on [0, y], ∀y ∈ [0, 1], u = v = Py = 0, on [x, 0] and [x, 1], ∀x ∈ [0, 1], ux = vx = 0, and Px = 0 on [1, y], ∀y ∈ [0, 1]. The equation (3.11), in partial finite differences, stands for (un − un−1 ) ∂ un un+1 − 2un + un−1 ∂ 2 un + − un − vn ν d2 ∂ y2 d ∂y Pn − Pn−1 − = 0, d vn+1 − 2vn + vn−1 ∂ 2 vn + d2 ∂ y2 ∂ Pn − = 0, ∂y

ν

Pn+1 − 2Pn + Pn−1 ∂ 2 Pn + d2 ∂ y2 ∂ un vn − vn−1 +2 = 0. ∂y d

− un

(3.12)

(vn − vn−1 ) ∂ vn − vn d ∂y (3.13)

(un − un−1 ) ∂ vn 2 + (un − un−1 ) + d2 ∂y (3.14)

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The Method of Lines and Duality Principles for Non-Convex Models

■

After linearizing such a system about U0 ,V0 , P0 and introducing the proximal formulation, for an appropriate non-negative real constant K,we get (un − (U0 )n−1 ) ∂ un un+1 − 2un + un−1 ∂ 2 un + − (U0 )n − (V0 )n d2 ∂ y2 d ∂y (P0 )n − (P0 )n−1 − − Kun + K(U0 )n = 0, (3.15) d

ν

(vn − (V0 )n−1 ) ∂ vn vn+1 − 2vn + vn−1 ∂ 2 vn + − (U0 )n − (V0 )n ν 2 2 d ∂y d ∂y ∂ (P0 )n − − Kvn + K(V0 )n = 0, (3.16) ∂y

((u)n+1 − (U0 )n ) ∂ (V0 )n 2 Pn+1 − 2Pn + Pn−1 ∂ 2 Pn + + (u − (U ) ) + n n+1 0 d2 ∂ y2 d2 ∂y ∂ (U0 )n vn+1 − (V0 )n +2 − KPn + K(P0 )n = 0. (3.17) ∂y d

At this point denoting ν = e1 , we define ∂ un (P0 )n − (P0 )n−1 d 2 ∂ 2 un 2 (un − (U0 )n−1 ) d , − (V0 )n − + (T1 )n = −(U0 )n d ∂y d e1 ∂ y2

(vn − (V0 )n−1 ) ∂ vn ∂ (P0 )n (T2 )n = −(U0 )n − (V0 )n − d ∂y ∂y

d 2 ∂ 2 vn 2 + d , e1 ∂ y2

and (T3 )n

((u)n+1 − (U0 )n ) 2 ∂ (V0 )n 2 2 = (un+1 − (U0 )n ) d + d d2 ∂y ∂ 2 Pn 2 ∂ (U0 )n vn+1 − (V0 )n d2 + d . (3.18) +2 ∂y d ∂ y2

Therefore, we may write un+1 − 2un + un−1 − K un

d2 + (T1 )n + ( f1 )n = 0, e1

Approximate Numerical Procedures for the Navier-Stokes System

where ( f1 )n = K(U0 )n

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43

d2 , e1

∀n ∈ {1, · · · , N − 1}. In particular for n = 1, we obtain u2 − 2u1 + u0 − K u1

d2 + (T1 )1 + ( f1 )1 = 0, e1

so that u1 = a1 u2 + b1 u0 + c1 (T1 )1 + (h1 )1 + (Er )1 , where

−1 d2 , a1 = 2 + K e1 b1 = a1 c1 = a1 (h1 )1 = a1 ( f1 )1 , (Er )1 = 0.

Similarly, for n = 2 we get u3 − 2u2 + u1 − Ku2

d2 + (T1 )2 + ( f1 )2 = 0, e2

so that u2 = a2 u3 + b2 u0 + c2 (T1 )2 + (h1 )2 + (Er )2 , where

−1 d2 , a2 = 2 + K − a1 e1 b2 = a2 b1 c2 = a2 (c1 + 1) (h1 )2 = a2 ((h1 )1 + ( f1 )2 ), (E1 )2 = a2 (c1 ((T1 )1 − (T1 )2 )).

Reasoning inductively, having un−1 = an−1 un + bn−1 u0 + cn−1 (T1 )n−1 + (h1 )n−1 + (Er )n−1 , we obtain un = an un+1 + bn u0 + cn (T1 )n + (h1 )n + (Er )n ,

(3.19)

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The Method of Lines and Duality Principles for Non-Convex Models

where

−1 d2 an = 2 + K − an−1 , e1 bn = an bn−1 cn = an (cn−1 + 1) (h1 )n = an ((h1 )n−1 + ( f1 )n ), (Er ) = an ((Er )n−1 + cn−1 ((T1 )n−1 − (T1 )n )),

∀n ∈ {1, · · · , N − 1}. Observe now that n = N − 1 we have uN−1 = uN , so that uN−1 ≈ aN−1 uN−1 + bN−1 u0 + cN−1 (T1 )N−1 + (h1 )N−1 aN−1 uN−1 + bN−1 u0 ∂ 2 uN−1 2 d +cN−1 ∂ y2 (un − (U0 )n−1 ) ∂ un (P0 )n − (P0 )n−1 d 2 +cN−1 −(U0 )n − (V0 )n − d ∂y d e1 +(h1 )N−1 (3.20) This last equation is a second order ODE in uN−1 which must be solved with the boundary conditions uN−1 (0) = uN−1 (1) = 0. Summarizing we have obtained uN−1 . Similarly, we may obtain vN−1 and PN−1 . Having uN−1 we may obtain uN−2 with n = N − 2 in equation (3.19) (neglecting (Er )N−2 .) Similarly, we may obtain vN−2 and PN−2 . Having uN−2 we may obtain uN−3 with n = N − 3 in equation (3.19) (neglecting (Er )N−3 .) Similarly, we may obtain vN−3 and PN−3 . And so on up to obtaining u1 , v1 and P1 . The next step is to replace {(U0 )n , (V0 )n , (P0 )n } by {un , vn , Pn } and repeat the process until an appropriate convergence criterion is satisfied. Here, we present a concerning software in MATLAB based in this last algorithm (with small changes and differences where we have set K = 155 and ν = 0.047).

Approximate Numerical Procedures for the Navier-Stokes System

******************************* clearall m8 = 500; d = 1/m8; m9 = 140; d1 = 1/m9; e1 = 0.05; K = 155.0; m2 = zeros(m9 − 1, m9 − 1); f or i = 2 : m9 − 2 m2(i, i) = −2.0; m2(i, i + 1) = 1.0; m2(i, i − 1) = 1.0; end; m2(1, 1) = −1.0; m2(1, 2) = 1.0; m2(m9 − 1, m9 − 1) = −1.0; m2(m9 − 1, m9 − 2) = 1.0; m22 = zeros(m9 − 1, m9 − 1); f or i = 2 : m9 − 2 m22(i, i) = −2.0; m22(i, i + 1) = 1.0; m22(i, i − 1) = 1.0; end; m22(1, 1) = −2.0; m22(1, 2) = 1.0; m22(m9 − 1, m9 − 1) = −2.0; m22(m9 − 1, m9 − 2) = 1.0; m1a = zeros(m9 − 1, m9 − 1); m1b = zeros(m9 − 1, m9 − 1); f or i = 1 : m9 − 2 m1a(i, i) = −1.0; m1a(i, i + 1) = 1.0; end; m1a(m9 − 1, m9 − 1) = −1.0; f or i = 2 : m9 − 1 m1b(i, i) = 1.0; m1b(i, i − 1) = −1.0; end; m1b(1, 1) = 1.0; m1 = (m1a + m1b)/2; Id = eye(m9 − 1); a(1) = 1/(2 + K ∗ d 2 /e1);

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45

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The Method of Lines and Duality Principles for Non-Convex Models

b(1) = 1/(2 + K ∗ d 2 /e1); c(1) = 1/(2 + K ∗ d 2 /e1); f or i = 2 : m8 − 1 a(i) = 1/(2 − a(i − 1) + K ∗ d 2 /e1); b(i) = a(i) ∗ b(i − 1); c(i) = (c(i − 1) + 1) ∗ a(i); end; f or i = 1 : m9 − 1 u5(i, 1) = 0.55 ∗ i ∗ d1 ∗ (1 − i ∗ d1); end; uo = u5; vo = zeros(m9 − 1, 1); po = 0.15 ∗ ones(m9 − 1, 1); f or i = 1 : m8 − 1 Uo(:, i) = 0.25 ∗ ones(m9 − 1, 1); Vo(:, i) = 0.05 ∗ ones(m9 − 1, 1); Po(:, i) = 0.05 ∗ ones(m9 − 1, 1); end; f or k7 = 1 : 1 e1 = e1 ∗ .94; b14 = 1.0; k1 = 1; k1max = 1000; while (b14 > 10−4.0 ) and (k1 < 1000) k1 = k1 + 1; a(1) = 1/(2 + K ∗ d 2 /e1); b(1) = a(1); c1(:, 1) = a(1) ∗ K ∗Uo(:, 1) ∗ d 2 /e1; c2(:, 1) = a(1) ∗ K ∗Vo(:, 1) ∗ d 2 /e1; c3(:, 1) = a(1) ∗ (K ∗ Po(:, 1) ∗ d 2 /e1); f or i = 2 : m8 − 1 a(i) = 1/(2 + K ∗ d 2 /e1 − a(i − 1)); b(i) = a(i) ∗ (b(i − 1)); c1(:, i) = a(i) ∗ (c1(:, i − 1) + K ∗Uo(:, i) ∗ d 2 /e1); c2(:, i) = a(i) ∗ (c2(:, i − 1) + K ∗Vo(:, i) ∗ d 2 /e1); c3(:, i) = a(i) ∗ (c3(:, i − 1) + K ∗ Po(:, i) ∗ d 2 /e1); end; i = 1; M50 = (Id − a(m8 − 1) ∗ Id − c(m8 − 1) ∗ m22/d12 ∗ d 2 ); z1 = b(m8 − 1) ∗ uo + c1(:, m8 − i) z1 = z1 + c(m8 − 1) ∗ (−Vo(:, m8 − i). ∗ (m1 ∗Uo(:, m8 − i)))/d1 ∗ d 2 /e1; M60 = (Id − a(m8 − 1) ∗ Id − c(m8 − 1) ∗ m22/d12 ∗ d 2 ); z2 = b(m8 − 1) ∗ vo + c2(:, m8 − i)

Approximate Numerical Procedures for the Navier-Stokes System

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47

z2 = z2 + c(m8 − 1) ∗ (−Vo(:, m8 − i)). ∗ (m1 ∗Vo(:, m8 − i))/d1 ∗ d 2 /e1; M70 = (Id − a(m8 − 1) ∗ Id − c(m8 − 1) ∗ m2/d12 ∗ d 2 ); z3 = b(m8 − 1) ∗ po; z3 = z3+c(m8−1)∗((m1/d1∗Vo(:, m8−i)).∗(m1/d1∗Vo(:, m8−i))∗d 2 ); z3 = z3 + c3(:, m8 − i); U(:, m8 − 1) = inv(M50) ∗ z1; V (:, m8 − 1) = inv(M60) ∗ z2; P(:, m8 − 1) = inv(M70) ∗ z3; f or i = 2 : m8 − 1 M50 = (Id − c(m8 − i) ∗ m22/d12 ∗ d 2 ); z1 = b(m8 − i) ∗ uo + a(m8 − i) ∗U(:, m8 − i + 1) z1 = z1 + c(m8 − i) ∗ (−U(:, m8 − i + 1). ∗ (Uo(:, m8 − i + 1) − Uo(:, m8 − i)) ∗ d/e1) z1 = z1 − c(m8 − i)(Po(:, m8 − i + 1) − Po(:, m8 − i)) ∗ d/e1; z1 = z1 + c1(:, m8 − i) + c(m8 − i) ∗ (−V (:, m8 − i + 1). ∗ (m1 ∗ Uo(:, m8 − i))/d1 ∗ d 2 )/e1; M60 = (Id − c(m8 − i) ∗ m22/d12 ∗ d 2 ); z2 = b(m8 − i) ∗ vo + a(m8 − i) ∗Vo(:, m8 − i + 1) z2 = z2 + c(m8 − i) ∗ (−U(:, m8 − i + 1). ∗ (Vo(:, m8 − i + 1) −Vo(:, m8 − i)) ∗ d/e1) z2 = z2 − c(m8 − i) ∗V (:, m8 − i + 1). ∗ (m1 ∗Vo(:, m8 − i)/d1 ∗ d 2 )/e1; z2 = z2 − c(m8 − i) ∗ (m1 ∗ Po(:, m8 − i)/d1 ∗ d 2 )/e1; z2 = z2 + c2(:, m8 − i); M70 = (Id − c(m8 − i) ∗ m2/d12 ∗ d 2 ); z3 = b(m8 − i) ∗ po + a(m8 − i) ∗ P(:, m8 − i + 1) z3 = z3 +c(m8 − i) ∗ ((Uo(:, m8 − i + 1) − Uo(:, m8 − i)). ∗ (Uo(:, m8 − i) − Uo(: , m8 − i))); z3 = z3 + c(m8 − i)(m1/d1 ∗Vo(:, m8 − i)). ∗ (m1/d1 ∗Vo(:, m8 − i)) ∗ d 2 ; z3 = z3 +2 ∗ (m1/d1 ∗ Uo(:, m8 − i)). ∗ (Vo(:, m8 − i + 1) − Vo(:, m8 − i)) ∗ d + c3(: , m8 − i); U(:, m8 − i) = inv(M50) ∗ z1; V (:, m8 − i) = inv(M60) ∗ z2; P(:, m8 − i) = inv(M70) ∗ z3; end; b14 = max(max(abs(U −Uo))); b14 Uo = U; Vo = V ; Po = P; k1 U(m9/2, 10)

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■ The Method of Lines and Duality Principles for Non-Convex Models

end; k7 end; f or i = 1 : m9 − 1 y(i) = i ∗ d1; end; f or i = 1 : m8 − 1 x(i) = i ∗ d; end; mesh(x, y,U); ********************************** For the field of velocities U, V and the pressure field P, please see Figures 3.1, 3.2 and 3.3, respectively.

Figure 3.1: Solution U(x, y) for the case ν = 0.047.

3.4

A software in MATHEMATICA related to the previous algorithm

In this section, we develop the solution for the Navier-Stokes system through the generalized method of lines, similar to the results presented in [18], but now in a Navier-Stokes system context.

Approximate Numerical Procedures for the Navier-Stokes System

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49

Figure 3.2: Solution V (x, y) for the case ν = 0.047.

Figure 3.3: Solution P(x, y) for the case ν = 0.047.

We present a software in MATHEMATICA for N = 10 lines for the case in which ν∇2 u − u∂x u − v∂y u − ∂x P = 0, in Ω, in Ω, ν∇2 v − u∂x v − v∂y v − ∂y P = 0, (3.21) 2 ∇ P + (∂x u)2 + (∂y v)2 + 2(∂y u)(∂x v) = 0, in Ω,

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The Method of Lines and Duality Principles for Non-Convex Models

We consider it in polar coordinates, with ν = e1 = 0.1, and with Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, and ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}. The boundary conditions are u = v = 0, P = 0.15 on ∂ Ω1 , u = u f [x], v = 0, P = 0.12 on ∂ Ω2 . From now and on, x stands for θ . We remark some changes have been made, concerning the original conception, to make it suitable through the software MATHEMATICA for such a Navier-Stokes system. We highlight, as K > 0 is larger, the related approximation is of a better quality. However if K > 0 is too much large, the converging process gets slower. Here the concerning software. ************************************************* 1. m8 = 10; Clear[t3,t4]; d = 1.0/m8; K = 4.0; e1 = 0.1; Uoo[x− ] = 0.0; Voo[x− ] = 0.0; Poo[x− ] = 0.15; 2. For[i = 1, i < m8 + 1, i + +, uo[i] = 0.05; vo[i] = 0.05; Po[i] = 0.05]; 3. For[k = 1, k < 80, k + +, (here we have fixed the number of iterations) Print[k]; a[1] = 1/(2.0 + K ∗ d 2 /e1); b[1] = a[1];

Approximate Numerical Procedures for the Navier-Stokes System

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b11[1] = a[1]; c1[1] = a[1] ∗ (K ∗ uo[1]) ∗ d 2 /e1; c2[1] = a[1] ∗ (K ∗ vo[1]) ∗ d 2 /e1; c3[1] = a[1] ∗ (K ∗ Po[1] + P1) ∗ d 2 /e1; 4. For[i = 2, i < m8, i + +, a[i] = 1/(2.0 + K ∗ d 2 /e1 − a[i − 1]); b[i] = a[i] ∗ (b[i − 1] + 1); b11[i] = a[i] ∗ b11[i − 1]; c1[i] = a[i] ∗ (c1[i − 1] + (K ∗ uo[i]) ∗ d 2 /e1); c2[i] = a[i] ∗ (c2[i − 1] + (K ∗ vo[i]) ∗ d 2 /e1); c3[i] = a[i] ∗ (c3[i − 1] + (K ∗ Po[i]) ∗ d 2 /e1)]; u[m8] = u f [x] ∗ t3; v[m8] = v f [x] ∗ t3; P[m8] = 0.12; d1 = 1.0; 5. For[i = 1, i < m8, i + +, Print[i]; t[m8 − i] = 1.0 + (m8 − i) ∗ d; Dxu = (uo[m8 − i + 1] − uo[m8 − i])/d ∗ f 1[x] ∗ t4 − D[uo[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; Dyu = (uo[m8 − i + 1] − uo[m8 − i])/d ∗ f 2[x] ∗ t4 + D[uo[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; Dxv = (vo[m8 − i + 1] − vo[m8 − i])/d ∗ f 1[x] ∗ t4 − D[vo[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; Dyv = (vo[m8 − i + 1] − vo[m8 − i])/d ∗ f 2[x] ∗ t4 + D[vo[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; DxP = (Po[m8 − i + 1] − Po[m8 − i])/d ∗ f 1[x] ∗ t4 − D[Po[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; DyP = (Po[m8 − i + 1] − Po[m8 − i])/d ∗ f 2[x] ∗ t4 + D[Po[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; T 1 = −(u[m8 − i + 1] ∗ Dxu + v[m8 − i + 1] ∗ Dyu + DxP) ∗ d 2 /e1 +(uo[m8 − i + 1] − uo[m8 − i])/d/t[m8 − i] ∗ d 2 + D[uo[m8 − i + 1], {x, 2}] /t[m8 − i]2 ∗ d 2 ; T 2 = −(u[m8 − i + 1] ∗ Dxv + v[m8 − i + 1] ∗ Dyv + DyP) ∗ d 2 /e1 + (vo[m8 − i + 1] − vo[m8 − i])/d/t[m8 − i] ∗ d 2 +D[vo[m8 − i + 1], {x, 2}]/t[m8 − i]2 ∗ d 2 ; T 3 = (Dxu2 + Dyv2 + 2 ∗ Dyu ∗ Dxv) ∗ d 2 +(Po[m8 − i + 1] − Po[m8 − i])/d/t[m8 − i] ∗ d 2 + D[Po[m8 − i + 1], {x, 2}]/t[m8 − i]2 ∗ d 2 ;

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6. A1 = a[m8 − i] ∗ u[m8 − i + 1] + b[m8 − i] ∗ T 1 + c1[m8 − i]; A1 = Expand[A1]; A1 = Series[A1, {t4, 0, 1}, {t3, 0, 2}, {u f [x], 0, 2}, {u f ′ [x], 0, 1}, {u f ′′ [x], 0, 1}, {u f ′′′ [x], 0, 0}, {u f ′′′′ [x], 0, 0}, {v f [x], 0, 1}, {v f ′ [x], 0, 1}, {v f ′′ [x], 0, 1}, {v f ′′′ [x], 0, 0}, {v f ′′′′ [x], 0, 0}, { f 1[x], 0, 1}, { f 2[x], 0, 1}, { f 1′ [x], 0, 0}, { f 2′ [x], 0, 0}, { f 1′′ [x], 0, 0}, { f 2′′ [x], 0, 0}]; A1 = Normal[A1]; u[m8 − i] = Expand[A1]; 7. A2 = a[m8 − i] ∗ v[m8 − i + 1] + b[m8 − i] ∗ T 2 + c2[m8 − i]; A2 = Expand[A2]; A2 = Series[A2, {t4, 0, 1}, {t3, 0, 2}, {u f [x], 0, 1}, {u f ′ [x], 0, 1}, {u f ′′ [x], 0, 1}, {u f ′′′ [x], 0, 0}, {u f ′′′′ [x], 0, 0}, {v f [x], 0, 2}, {v f ′ [x], 0, 1}, {v f ′′ [x], 0, 1}, {v f ′′′ [x], 0, 0}, {v f ′′′′ [x], 0, 0}, { f 1[x], 0, 1}, { f 2[x], 0, 1}, { f 1′ [x], 0, 0}, { f 2′ [x], 0, 0}, { f 1′′ [x], 0, 0}, { f 2′′ [x], 0, 0}]; A2 = Normal[A2]; v[m8 − i] = Expand[A2]; 8. A3 = a[m8 − i] ∗ P[m8 − i + 1] + b[m8 − i] ∗ T 3 + c3[m8 − i] + b11[m8 − i] ∗ Poo[x]; A3 = Expand[A3]; A3 = Series[A3, {t4, 0, 2}, {t3, 0, 2}, {u f [x], 0, 1}, {u f ′ [x], 0, 1}, {u f ′′ [x], 0, 0}, {u f ′′′ [x], 0, 0}, {u f ′′′′ [x], 0, 0}, {v f [x], 0, 1}, {v f ′ [x], 0, 1}, {v f ′′ [x], 0, 0}, {v f ′′′ [x], 0, 0}, {v f ′′′′ [x], 0, 0}, { f 1[x], 0, 1}, { f 2[x], 0, 1}, { f 1′ [x], 0, 0}, { f 2′ [x], 0, 0}, { f 1′′ [x], 0, 0}, { f 2′′ [x], 0, 0}]; A3 = Normal[A3]; P[m8 − i] = Expand[A3]]; 9. For[i = 1, i < m8 + 1, i + +, uo[i] = u[i]; vo[i] = v[i]; Po[i] = P[i]]; d1 = 1.0;

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10. Print[Expand[U[m8/2]]]] 11. For[i = 1, i < m8 , i + +, Print[“u[”, i, “] = ”, u[i][x]]] Here we present the related line expressions obtained for the lines n = 1, n = 5 and n=9 of a total of N = 10 lines. 1. Line n = 1 u[1] =

1.58658 ∗ 10−9 + 0.019216 f1 [x] + 3.68259 ∗ 10−11 f2 [x] + 0.132814u f [x] +2.28545 ∗ 10−8 f1 [x]u f [x] − 1.69037 ∗ 10−8 f2 [x]u f [x] − 0.263288 f1 [x]u f [x]2 −6.02845 ∗ 10−9 f1 [x](u′f )[x] + 6.02845 ∗ 10−9 f2 [x](u′f )[x] +0.104284 f2 [x]u f [x](u′f ])[x] + 0.0127544(u′′f )[x] +9.39885 ∗ 10−9 f1 [x](u′′f )[x] − 5.26303 ∗ 10−9 f2 [x](u′′f )[x] −0.0340276 f1 [x]u f [x](u′′f )[x] + 0.0239544 f2 [x](u′f )[x](u′′f )[x]

(3.22)

2. Line n = 5 u[5]

=

4.25933 ∗ 10−9 + 0.0436523 f1 [x] + 9.88625 ∗ 10−11 f2 [x] + 0.572969u f [x] +6.87985 ∗ 10−8 f1 [x]u f [x] − 4.40534 ∗ 10−8 f2 [x]u f [x] − 0.765222 f1 [x]u f [x]2 −1.61319 ∗ 10−8 f1 [x](u′f )[x] + 1.61319 ∗ 10−8 f2 [x](u′f )[x] +0.363471 f2 [x]u f [x](u′f )[x] + 0.0333685(u′′f )[x] +2.39576 ∗ 10−8 f1 [x](u′′f )[x] − 1.27491 ∗ 10−8 f2 [x](u′′f )[x] −0.0342544 f1 [x]u f [x](u′′f )[x] + 0.0509889 f2 [x](u′f )[x](u′′f )[x]

(3.23)

3. Line n = 9 u[9]

=

1.15848 ∗ 10−9 + 0.0136828 f1 [x] + 2.68892 ∗ 10−11 f2 [x] + 0.922534u f [x] +2.16498 ∗ 10−8 f1 [x]u f [x] − 1.16065 ∗ 10−8 f2 [x]u f [x] − 0.278966 f1 [x]u f [x]2 −4.25263 ∗ 10−9 f1 [x](u f ′ )[x] + 4.25263 ∗ 10−9 f2 [x](u′f )[x] +0.154642 f2 [x]u f [x](u′f )[x] + 0.0110114(u′′f )[x] +6.13523 ∗ 10−9 f1 [x](u′′f )[x] − 3.23081 ∗ 10−9 f2 [x](u′′f )[x] +0.0146222 f1 [x]u f [x](u′′f )[x] + 0.0090088 f2 [x](u′f )[x](u′′f )[x]

(3.24)

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The Method of Lines and Duality Principles for Non-Convex Models

The software and numerical results for a more specific example

In this section, we present numerical results for the same Navier-Stokes system and domain as in the previous one, but now with different boundary conditions. In this example, we set ν = 0.1 and the boundary conditions are u = v = 0, P = 0.15 on ∂ Ω1 , u = −1.0 sin[x], v = 1.0 cos[x], P = 0.12 on ∂ Ω2 . Here the concerning software: ************************************************* 1. m8 = 10; Clear[t3,t4]; d = 1.0/m8; K = 4.0; e1 = 0.1; Uoo[x− ] = 0.0; Voo[x− ] = 0.0; Poo[x− ] = 0.15; 2. For[i = 1, i < m8 + 1, i + +, uo[i] = 0.05; vo[i] = 0.05; Po[i] = 0.05]; f 1[x− ] = Cos[x]; f 2[x− ] = Sin[x]; 3. For[k = 1, k < 80, k + +, (here we have fixed the number of iterations) Print[k]; a[1] = 1/(2.0 + K ∗ d 2 /e1); b[1] = a[1]; b11[1] = a[1]; c1[1] = a[1] ∗ (K ∗ uo[1]) ∗ d 2 /e1; c2[1] = a[1] ∗ (K ∗ vo[1]) ∗ d 2 /e1; c3[1] = a[1] ∗ (K ∗ Po[1] + P1) ∗ d 2 /e1;

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4. For[i = 2, i < m8, i + +, a[i] = 1/(2.0 + K ∗ d 2 /e1 − a[i − 1]); b[i] = a[i] ∗ (b[i − 1] + 1); b11[i] = a[i] ∗ b11[i − 1]; c1[i] = a[i] ∗ (c1[i − 1] + (K ∗ uo[i]) ∗ d 2 /e1); c2[i] = a[i] ∗ (c2[i − 1] + (K ∗ vo[i]) ∗ d 2 /e1); c3[i] = a[i] ∗ (c3[i − 1] + (K ∗ Po[i]) ∗ d 2 /e1)]; u f [x− ] = −1.0 ∗ Sin[x]; v f [x] = 1.0 ∗Cos[x]; u[m8] = u f [x] ∗ t3; v[m8] = v f [x] ∗ t3; P[m8] = 0.12; 5. For[i = 1, i < m8, i + +, Print[i]; t[m8 − i] = 1.0 + (m8 − i) ∗ d; Dxu = (uo[m8 − i + 1] − uo[m8 − i])/d ∗ f 1[x] ∗ t4 − D[uo[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; Dyu = (uo[m8 − i + 1] − uo[m8 − i])/d ∗ f 2[x] ∗ t4 + D[uo[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; Dxv = (vo[m8 − i + 1] − vo[m8 − i])/d ∗ f 1[x] ∗ t4 − D[vo[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; Dyv = (vo[m8 − i + 1] − vo[m8 − i])/d ∗ f 2[x] ∗ t4 + D[vo[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; DxP = (Po[m8 − i + 1] − Po[m8 − i])/d ∗ f 1[x] ∗ t4 − D[Po[m8 − i], x] ∗ f 2[x]/t[m8 − i] ∗ t4; DyP = (Po[m8 − i + 1] − Po[m8 − i])/d ∗ f 2[x] ∗ t4 + D[Po[m8 − i], x] ∗ f 1[x]/t[m8 − i] ∗ t4; T 1 = −(u[m8 − i + 1] ∗ Dxu + v[m8 − i + 1] ∗ Dyu + DxP) ∗ d 2 /e1 +(uo[m8 − i + 1] − uo[m8 − i])/d/t[m8 − i] ∗ d 2 +D[uo[m8 − i + 1], {x, 2}]/t[m8 − i]2 ∗ d 2 ; T 2 = −(u[m8 − i + 1] ∗ Dxv + v[m8 − i + 1] ∗ Dyv + DyP) ∗ d 2 /e1 +(vo[m8 − i + 1] − vo[m8 − i])/d/t[m8 − i] ∗ d 2 + D[vo[m8 − i + 1], {x, 2}] /t[m8 − i]2 ∗ d 2 ; T 3 = (Dxu2 + Dyv2 + 2 ∗ Dyu ∗ Dxv) ∗ d 2

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+(Po[m8 − i + 1] − Po[m8 − i])/d/t[m8 − i] ∗ d 2 + D[Po[m8 − i + 1], {x, 2}] /t[m8 − i]2 ∗ d 2 ; A1 = a[m8 − i] ∗ u[m8 − i + 1] + b[m8 − i] ∗ T 1 + c1[m8 − i]; A1 = Expand[A1]; A1 = Series[A1, {t4, 0, 1}, {t3, 0, 2}, {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A1 = Normal[A1]; u[m8 − i] = Expand[A1]; A2 = a[m8 − i] ∗ v[m8 − i + 1] + b[m8 − i] ∗ T 2 + c2[m8 − i]; A2 = Expand[A2]; A2 = Series[A2, {t4, 0, 1}, {t3, 0, 2}, {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A2 = Normal[A2]; v[m8 − i] = Expand[A2]; A3 = a[m8 − i] ∗ P[m8 − i + 1] + b[m8 − i] ∗ T 3 + c3[m8 − i] + b11[m8 − i] ∗ Poo[x]; A3 = Expand[A3]; A3 = Series[A3, {t4, 0, 2}, {t3, 0, 2}, {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A3 = Normal[A3]; P[m8 − i] = Expand[A3]]; 6. For[i = 1, i < m8 + 1, i + +, uo[i] = u[i]; vo[i] = v[i]; Po[i] = P[i]]; Print[Expand[P[m8/2]]]] 7. For[i = 1, i < m8 , i + +, Print[“u[”, i, “] = ”, u[i][x]]] Here the corresponding line expressions for N = 10 lines 1. Line n = 1 u[1]

=

1.445 ∗ 10−10 + 0.0183921Cos[x] + 1.01021 ∗ 10−9Cos[x]2 − 0.120676Sin[x] +3.62358 ∗ 10−10Cos[x]Sin[x] + 1.37257 ∗ 10−9 Sin[x]2 +0.0620534Cos[x]Sin[x]2

(3.25)

2. Line n = 2 u[2] =

2.60007 ∗ 10−10 + 0.0307976Cos[x] + 1.81088 ∗ 10−9Cos[x]2 − 0.233061Sin[x] +6.53242 ∗ 10−10Cos[x]Sin[x] + 2.46412 ∗ 10−9 Sin[x]2 +0.123121Cos[x]Sin[x]2

(3.26)

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3. Line n = 3 u[3] =

3.40796 ∗ 10−10 + 0.0384482Cos[x] + 2.36167 ∗ 10−9Cos[x]2 − 0.339657Sin[x] +8.5759 ∗ 10−10Cos[x]Sin[x] + 3.21926 ∗ 10−9 Sin[x]2 +0.180891Cos[x]Sin[x]2

(3.27)

4. Line n = 4 u[4] =

3.83612 ∗ 10−10 + 0.0420843Cos[x] + 2.64262 ∗ 10−9Cos[x]2 − 0.441913Sin[x] +9.66336 ∗ 10−10Cos[x]Sin[x] + 3.60895 ∗ 10−9 Sin[x]2 +0.230559Cos[x]Sin[x]2

(3.28)

5. Line n = 5 u[5] = 3.87923 ∗ 10−10 + 0.0421948Cos[x] + 2.65457 ∗ 10−9Cos[x]2 − 0.540729Sin[x] +9.77606 ∗ 10−10Cos[x]Sin[x] + 3.63217 ∗ 10−9 Sin[x]2 +0.266239Cos[x]Sin[x]2

(3.29)

6. Line n = 6 u[6]

=

3.56064 ∗ 10−10 + 0.0391334Cos[x] + 2.419 ∗ 10−9Cos[x]2 − 0.636718Sin[x] +8.97185 ∗ 10−10Cos[x]Sin[x] + 3.31618 ∗ 10−9 Sin[x]2 +0.281514Cos[x]Sin[x]2

(3.30)

7. Line n = 7 u[7]

= 2.93128 ∗ 10−10 + 0.033175Cos[x] + 1.97614 ∗ 10−9Cos[x]2 − 0.730328Sin[x] +7.38127 ∗ 10−10Cos[x]Sin[x] + 2.71426 ∗ 10−9 Sin[x]2 +0.269642Cos[x]Sin[x]2

(3.31)

8. Line n = 8 u[8]

= 2.0656 ∗ 10−10 + 0.0245445Cos[x] + 1.38127 ∗ 10−9Cos[x]2 − 0.821902Sin[x] +5.19585 ∗ 10−10Cos[x]Sin[x] + 1.90085 ∗ 10−9 Sin[x]2 +0.22362Cos[x]Sin[x]2

(3.32)

9. Line n = 9 u[9] = 1.05509 ∗ 10−10 + 0.0134316Cos[x] + 6.99554 ∗ 10−10Cos[x]2 − 0.911718Sin[x] +2.65019 ∗ 10−10Cos[x]Sin[x] + 9.64573 ∗ 10−10 Sin[x]2 +0.13622Cos[x]Sin[x]2

(3.33)

Here we present the related plots for the Lines n = 2, n = 4, n = 6 and n = 8 of a total of N = 10 lines. For each line we set N = 500 nodes on the interval [0, 2π], so that the units in x are 2π/500, where again x stands for θ . For such lines, please see Figures 3.4, 3.5, 3.6 and 3.7, respectively.

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0.3

0.2

0.1

0

-0.1

-0.2

-0.3 0

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Figure 3.4: Solution u2 (x) for the line n = 2, for the case ν = 0.1.

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0

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Figure 3.5: Solution u4 (x) for the line n = 4, for the case ν = 0.1.

3.6

Numerical results through the original conception of the generalized method of lines for the NavierStokes system

In this section, we develop the solution for the Navier-Stokes system through the generalized method of lines, as originally introduced in [22], with further developments in [17].

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0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

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Figure 3.6: Solution u6 (x) for the line n = 6, for the case ν = 0.1.

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

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Figure 3.7: Solution u8 (x) for the line n = 8, for the case ν = 0.1.

We present a software in MATHEMATICA for N = 10 lines for the case in which ν∇2 u − u∂x u − v∂y u − ∂x P = 0, in Ω, ν∇2 v − u∂x v − v∂y v − ∂y P = 0, in Ω, (3.34) 2 ∇ P + (∂x u)2 + (∂y v)2 + 2(∂y u)(∂x v) = 0, in Ω.

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Such a software refers to an algorithm presented in Chapter 27, in [13], in polar coordinates, with ν = 1.0, and with Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, and ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}. The boundary conditions are u = v = 0, P = 0.15 on ∂ Ω1 , u = −1.0 sin(θ ), v = 1.0 cos(θ ), P = 0.10 on ∂ Ω2 . We remark some changes have been made, concerning the original conception, to make it suitable through the software MATHEMATICA for such a Navier-Stokes system. We highlight the nature of this approximation is qualitative. Here the concerning software in MATHEMATICA. ***************************************** 1. m8 = 10; 2. Clear[z1 , z2 , z3 , u1 , u2 , P, b1 , b2 , b3 , a1 , a2 , a3 ]; 3. Clear[Pf ,t, a11 , a12 , a13 , b11 , b12 , b13 ,t3 ]; 4. d = 1.0/m8; 5. e1 = 1.0; 6. a1 = 0.0; 7. a2 = 0.0; 8. a3 = 0.15; 9. For[i = 1, i < m8, i + +, Print[i]; Clear[b1 , b2 , b3 , u1 , u2 , P]; b1 [x− ] = u1 [i + 1][x]; b2 [x− ] = u2 [i + 1][x]; b3 [x− ] = P[i + 1][x]; t[i] = 1 + i ∗ d;

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du1x = Cos[x] ∗ (b1 [x] − a1 )/d ∗ t3 − 1/t[i] ∗ Sin[x] ∗ D[b1 [x], x] ∗ t3 ; du1y = Sin[x] ∗ (b1 [x] − a1 )/d ∗ t3 + 1/t[i] ∗Cos[x] ∗ D[b1 [x], x] ∗ t3 ; du2x = Cos[x] ∗ (b2 [x] − a2 )/d ∗ t3 − 1/t[i] ∗ Sin[x] ∗ D[b2 [x], x] ∗ t3 ; du2y = Sin[x] ∗ (b2 [x] − a2 )/d ∗ t3 + 1/t[i] ∗Cos[x] ∗ D[b2 [x], x] ∗ t3 ; dPx = Cos[x] ∗ (b3 [x] − a3 )/d ∗ t3 − 1/t[i] ∗ Sin[x] ∗ D[b3 [x], x] ∗ t3 ; dPy = Sin[x] ∗ (b3 [x] − a3 )/d ∗ t3 + 1/t[i] ∗Cos[x] ∗ D[b3 [x], x] ∗ t3 ; 10. For[k = 1, k < 6, k + +, (in this example, we have fixed a relatively small number of iterations ) Print[k]; z1 = (u1 [i + 1][x] + b1 [x] + a1 + 1/t[i] ∗ (b1 [x] − a1 ) ∗ d + 1/t[i]2 ∗ D[b1 [x], x, 2] ∗ d 2 −(b1 [x] ∗ du1x + b2 [x] ∗ du1y) ∗ d 2 /e1 − dPx ∗ d 2 /e1 )/3.0; z2 = (u2 [i + 1][x] + b2 [x] + a2 + 1/t[i] ∗ (b2 [x] − a2 ) ∗ d + 1/t[i]2 ∗ D[b2 [x], x, 2] ∗ d 2 −(b1 [x] ∗ du2x + b2 [x] ∗ du2y) ∗ d 2 /e1 − dPy ∗ d 2 /e1 )/3.0; z3 = (P[i + 1][x] + b3 [x] + a3 + 1/t[i] ∗ (b3 [x] − a3 ) ∗ d + 1/t[i]2 ∗ D[b3 [x], x, 2] ∗ d 2 +(du1x ∗ du1x + du2y ∗ du2y + 2.0 ∗ du1y ∗ du2x) ∗ d 2 )/3.0; 11. z1 = Series[z1 , {u1 [i + 1][x], 0, 2}, {u1 [i + 1]′ [x], 0, 1}, {u1 [i + 1]′′ [x], 0, 1}, {u1 [i + 1]′′′ [x], 0, 0}, {u1 [i + 1]′′′′ [x], 0, 0}, {u2 [i + 1][x], 0, 1}, {u2 [i + 1]′ [x], 0, 0}, {u2 [i + 1]′′ [x], 0, 0}, {u2 [i + 1]′′′ [x], 0, 0}, {u2 [i + 1]′′′′ [x], 0, 0}, {P[i + 1][x], 0, 1}, {P[i + 1]′ [x], 0, 0}, {P[i + 1]′′ [x], 0, 0}, {P[i + 1]′′′ [x], 0, 0}, {P[i + 1]′′′′ [x], 0, 0}, {Sin[x], 0, 1}, {Cos[x], 0, 1}]; z1 = Normal[z1 ]; z1 = Expand[z1 ]; Print[z1 ]; 12. z2 = Series[z2 , {u1 [i + 1][x], 0, 1}, {u1 [i + 1]′ [x], 0, 1}, {u1 [i + 1]′′ [x], 0, 1}, {u1 [i + 1]′′′ [x], 0, 0}, {u1 [i + 1]′′′′ [x], 0, 0}, {u2 [i + 1][x], 0, 2}, {u2 [i + 1]′ [x], 0, 0}, {u2 [i + 1]′′ [x], 0, 0}, {u2 [i + 1]′′′ [x], 0, 0}, {u2 [i + 1]′′′′ [x], 0, 0}, {P[i + 1][x], 0, 1}, {P[i + 1]′ [x], 0, 0}, {P[i + 1]′′ [x], 0, 0}, {P[i + 1]′′′ [x], 0, 0}, {P[i + 1]′′′′ [x], 0, 0}, {Sin[x], 0, 1}, {Cos[x], 0, 1}];

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z2 = Normal[z2 ]; z2 = Expand[z2 ]; Print[z2 ]; 13. z3 = Series[z3 , {u1 [i + 1][x], 0, 2}, {u1 [i + 1]′ [x], 0, 1}, {u1 [i + 1]′′ [x], 0, 1}, {u1 [i + 1]′′′ [x], 0, 0}, {u1 [i + 1]′′′′ [x], 0, 0}, {u2 [i + 1][x], 0, 2}, {u2 [i + 1]′ [x], 0, 1}, {u2 [i + 1]′′ [x], 0, 0}, {u2 [i + 1]′′′ [x], 0, 0}, {u2 [i + 1]′′′′ [x], 0, 0}, {P[i + 1][x], 0, 1}, {P[i + 1]′ [x], 0, 1}, {P[i + 1]′′ [x], 0, 0}, {P[i + 1]′′′ [x], 0, 0}, {P[i + 1]′′′′ [x], 0, 0}, {Sin[x], 0, 1}, {Cos[x], 0, 1}]; z3 = Normal[z3 ]; z3 = Expand[z3 ]; Print[z3 ]; 14. b1 [x− ] = z1 ; b2 [x− ] = z2 ; b3 [x− ] = z3 ; b11 = z1 ; b12 = z2 ; b13 = z3 ; ]; 15. a11 [i] = b11 ; a12 [i] = b12 ; a13 [i] = b13 ; Print[a11 [i]]; Clear[b1 , b2 , b3 ]; u1 [i + 1][x− ] = b1 [x]; u2 [i + 1][x− ] = b2 [x]; P[i + 1][x− ] = b3 [x]; a1 = Series[b11 , {t3 , 0, 1}; {b1 [x], 0, 1}, {b2 [x], 0, 1}, {b3 [x], 0, 1}, {b′1 [x], 0, 0}, {b′2 [x], 0, 0}, {b′3 [x], 0, 0}, {b′′1 [x], 0, 0}, {b′′2 [x], 0, 0}, {b′′3 [x], 0, 0}]; a1 = Normal[a1 ]; a1 = Expand[a1 ]; a2 = Series[b12 , {t3 , 0, 1}; {b1 [x], 0, 1}, {b2 [x], 0, 1}, {b3 [x], 0, 1},

Approximate Numerical Procedures for the Navier-Stokes System

{b′1 [x], 0, 0}, {b′2 [x], 0, 0}, {b′3 [x], 0, 0}, {b′′1 [x], 0, 0}, {b′′2 [x], 0, 0}, {b′′3 [x], 0, 0}]; a2 = Normal[a2 ]; a2 = Expand[a2 ]; a3 = Series[b13 , {t3 , 0, 1}; {b1 [x], 0, 1}, {b2 [x], 0, 1}, {b3 [x], 0, 1}, {b′1 [x], 0, 0}, {b′2 [x], 0, 0}, {b′3 [x], 0, 0}, {b′′1 [x], 0, 0}, {b′′2 [x], 0, 0}, {b′′3 [x], 0, 0}]; a3 = Normal[a3 ]; a3 = Expand[a3 ]; 16. b1 [x] = −1.0 ∗ Sin[x]; b2 [x] = 1.0 ∗Cos[x]; b3 [x] = 0.10; 17. For[i = 1, i < m8, i + +, A11 = a11 [m8 − i]; A11 = Series[A11 , {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A11 = Normal[A11 ]; A11 = Expand[A11 ]; A12 = a12 [m8 − i]; A12 = Series[A12 , {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A12 = Normal[A12 ]; A12 = Expand[A12 ]; A13 = a13 [m8 − i]; A13 = Series[A13 , {Sin[x], 0, 2}, {Cos[x], 0, 2}]; A13 = Normal[A13 ]; A13 = Expand[A13 ]; 18. u1 [m8 − i][x− ] = A11 ; u2 [m8 − i][x− ] = A12 ; P[m8 − i][x− ] = Expand[A13 ]; 19. t3 = 1.0; 20. Print[“u1 [”, m8 − i, “] = ”, A11 ]; Clear[t3 ]; b1 [x] = A11 ; b2 [x] = A12 ; b3 [x] = A13 ; ];

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The Method of Lines and Duality Principles for Non-Convex Models

21. t3 = 1.0; 22. For[i = 1, i < m8 , i + +, Print[“u1 [”, i, “] = ”, u1 [i][x]]] *************************************************** Here the line expressions for the field of velocity u = {u1 [n](x)}, where again we emphasize N = 10 lines and ν = e1 = 1.0: 1. u1 [1](x)

=

0.0044548Cos[x] − 0.174091Sin[x] + 0.00041254Cos[x]2 Sin[x] +0.0260471Cos[x]Sin[x]2 − 0.000188598Cos[x]2 Sin[x]2

(3.35)

2. u1 [2](x)

=

0.00680614Cos[x] − 0.331937Sin[x] + 0.000676383Cos[x]2 Sin[x] +0.0501544Cos[x]Sin[x]2 − 0.000176433Cos[x]2 Sin[x]2

(3.36)

3. u1 [3](x)

=

0.00775103Cos[x] − 0.470361Sin[x] + 0.000863068Cos[x]2 Sin[x] +0.0682792Cos[x]Sin[x]2 − 0.000121656Cos[x]2 Sin[x]2

(3.37)

4. u1 [4](x)

=

0.00771379Cos[x] − 0.589227Sin[x] + 0.000994973Cos[x]2 Sin[x] +0.0781784Cos[x]Sin[x]2 − 0.00006958Cos[x]2 Sin[x]2

(3.38)

5. u1 [5](x)

=

0.00701567Cos[x] − 0.690152Sin[x] + 0.00106158Cos[x]2 Sin[x] +0.0796091Cos[x]Sin[x]2 − 0.0000330485Cos[x]2 Sin[x]2

(3.39)

6. u1 [6](x)

=

0.00589597Cos[x] − 0.775316Sin[x] + 0.00104499Cos[x]2 Sin[x] +0.0734277Cos[x]Sin[x]2 − 0.0000121648Cos[x]2 Sin[x]2

(3.40)

7. u1 [7](x)

=

0.00452865Cos[x] − 0.846947Sin[x] + 0.000931782Cos[x]2 Sin[x] +0.0609739Cos[x]Sin[x]2 − 2.74137 ∗ 10−6Cos[x]2 Sin[x]2

(3.41)

Approximate Numerical Procedures for the Navier-Stokes System

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65

8. u1 [8](x)

=

0.00303746Cos[x] − 0.907103Sin[x] + 0.000716865Cos[x]2 Sin[x] +0.0437018Cos[x]Sin[x]2

(3.42)

9. u1 [9](x)

=

0.00150848Cos[x] − 0.957599Sin[x] + 0.000403216Cos[x]2 Sin[x] +0.0229802Cos[x]Sin[x]2

3.7

(3.43)

Conclusion

In this chapter, we develop solutions for examples concerning the twodimensional, time-independent, and incompressible Navier-Stokes system through the generalized method of lines. We also obtain the appropriate boundary conditions for an equivalent elliptic system. Finally, the extension of such results to R3 , compressible and time dependent cases is planned for a future work.

Chapter 4

An Approximate Numerical Method for Ordinary Differential Equation Systems with Applications to a Flight Mechanics Model

4.1

Introduction

This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the generalized method of lines, originally published in the referenced books of 2011 and 2014 [22, 12], respectively. Finally, in the last section, we apply the method to a model in flight mechanics.

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Consider the first order system of ordinary differential equations given by du j = f j ({ul }), on [0,t f ] ∀ j ∈ {1, · · · , 4}, dt with the boundary conditions u1 (0) = 0, u2 (0), u4 (0) = 0, u4 (t f ) = u f . Here u = {ul } ∈ W 1,2 ([0,t f ], R4 ) and f j are functions at least of C1 class, ∀ j ∈ {1, 2, 3, 4}. Our proposed method is iterative so that we choose an starting solution denoted by u. ˜ At this point, we define the number of nodes on [0,t f ] by N and set d = t f /N. Similarly to a proximal approach, we propose the following algorithm (in a similar fashion as those found in [22, 12, 13]). 1. Choose u˜ ∈ W 1,2 ([0,t f ], R4 ). 2. Solve the equation system d u˜ j du j − (K − 1) = f j ({ul }), on [0,t f ] ∀ j ∈ {1, · · · , 4}, dt dt with the boundary conditions K

(4.1)

u1 (0) = 0, u2 (0), u4 (0) = 0, u4 (t f ) = u f . 3. Replace u˜ by u and go to item (2) up to the satisfaction of an appropriate convergence criterion. In finite differences, observe that the system may be approximated by (u j )n − (u j )n−1 =

K −1 d ((u˜ j )n − (u˜ j )n−1 ) + f j ({(ul )n−1 }) K K

In particular, for n = 1, we get (u j )1 = (u j )0 + f j ({(ul )1 })

d K −1 + ((u˜ j )1 − (u˜ j )0 ) + (E j )1 , K K

where (E j )1 = [ f j ({(ul )0 }) − f j ({(ul )1 })]

d ≈O K

d2 K

.

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The Method of Lines and Duality Principles for Non-Convex Models

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Similarly, for n = 2 we have (u j )2 − (u j )0 = [(u j )2 − (u j )1 ] + [(u j )1 − (u j )0 ] d K −1 ((u˜ j )2 − (u˜ j )1 ) = f j ({(ul )1 }) + K K d K −1 + f j ({(ul )1 }) + ((u˜ j )1 − (u˜ j )0 ) + (E j )1 K K d d d = 2 f j ({(ul )1 }) − 2 f j ({(ul )2 }) + 2 f j ({(ul )2 }) K K K K −1 + ((u˜ j )2 − (u˜ j )0 ) + (E j )1 , (4.2) K Summarizing (u j )2 = (u j )0 + 2 f j ({(ul )2 })

d K −1 + ((u˜ j )2 − (u˜ j )0 ) + (E j )2 , K K

where (E j )2 = (E j )1 + ( f j ({(ul )1 }) − f j ({(ul )2 }))

2d . K

Reasoning inductively, for all 1 ≤ k ≤ N, we obtain (u j )k = (u j )0 + f j ({(ul )k })

kd K − 1 + ((u˜ j )k − (u˜ j )0 ) + (E j )k , K K

where, (E j )k

d K 2 k (k + k)d 2 d , (4.3) = ∑ m[ f j ({(ul )m−1 }) − f j ({(ul )m })] ≈ O K 2K m=1 = (E j )k−1 + k( f j ({ul }k−1 ) − f j ({ul }k )

∀ j ∈ {1, · · · , 4}. In particular, for k = N, for K > 0 sufficiently big, we obtain Nd K − 1 + ((u˜ j )N − (u˜ j )0 ) + (E j )N K K Nd K − 1 ≈ (u j )0 + f j ({(ul )N }) + ((u˜ j )N − (u˜ j )0 ), ∀ j ∈ {1, 2, 3, 4}. (4.4) K K

(u j )N = (u j )0 + f j ({(ul )N })

In such a system we have 4 equations suitable to find the unknown variables (u0 )3 , (uN )1 , (uN )2 , (uN )3 , considering that (uN )4 = (u4 ) f is known. Through the system (4.4), we obtain uN through the Newton’s Method for a system of only the 4 indicated variables.

An Approximate Procedure for Ordinary Differential Equation Systems

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69

Having uN , we obtain uN−1 through the equations (u j )N − (u j )N−1 ≈ f j ({(ul )N−1 })

d K −1 + ((u˜ j )N ) − (u˜ j )N−1 ), K K

also through the Newton’s Method. Similarly, having uN−1 we obtain uN−2 through the system (u j )N−1 − (u j )N−2 ≈ f j ({(ul )N−2 })

d K −1 + ((u˜ j )N−1 ) − (u˜ j )N−2 ), K K

and so on up to finding u1 . Having calculated u, we replace u˜ by u and repeat the process up to an appropriate convergence criterion is satisfied. The problem is then approximately solved.

4.2

Applications to a flight mechanics model

We present numerical results for the following system of equations, which models the in plane climbing motion of an airplane (please, see more details in [83]). h˙ = V sin γ, γ˙ = 1 (F sin[a + aF ] + L) − g cos γ, mfV V (4.5) 1 ˙ V = (F cos[a + a ]) − D) − g sin γ F mf x˙ = V cos γ, with the boundary conditions, h(0) = h0 , V (0) = V0 x(0) = x0 h(t f ) = h f ,

(4.6)

where t f = 505s, h is the airplane altitude, V is its speed, γ is the angle between its velocity and the horizontal axis, and finally x denotes the horizontal coordinate position. For numerical purposes, we assume (Air bus 320) m f = 120, 000Kg, S f = 260m2 , a = 0.138 rad, g = 9.8m/s2 , ρ(h) = 1.225(1 − 0.0065h/288.15)4.225 Kg/m3 , aF = 0.0175, (CL )a = 5 (CD )0 = 0.0175,

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K1 = 0.0, K2 = 0.06 CD = (CD )0 + K1 1CL + K2CL2 , CL = (CL )0 + (CL )a a, 1 L = ρ(h)V 2CL S f , 2 1 D = ρ(h)V 2CD S f , 2 F = 240000 and where units refer to the International System. To simplify the analysis, we redefine the variables as below indicated: h = u1 , γ = u2 (4.7) V = u3 x = u4 . Thus, denoting u = (u1 , u2 , u3 , u4 ) ∈ U = W 1,2 ([0,t f ]; R4 ), the system above indicated may be expressed by u˙1 = f1 (u) u˙2 = f2 (u) (4.8) u˙3 = f3 (u) u˙4 = f4 (u), where, f1 (u) = u3 sin(u2 ), f2 (u) = 1 (F sin[a + aF ] + L(u)) − g cos(u2 ), m f u3 u3 f3 (u) = m1f (F cos[a + aF ] − D(u)) − g sin(u2 ) f4 (u) = u3 cos(u2 ).

We solve this last system for the following boundary conditions: h(0) = 0 m, V (0) = 120m/s, x(0) = 0 m, h(t f ) = 11000 m.

(4.9)

(4.10)

An Approximate Procedure for Ordinary Differential Equation Systems

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71

12000

10000

8000

6000

4000

2000

0 0

500

1000

1500

2000

2500

3000

Figure 4.1: The solution h (in m) for t f = 505s.

0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0

500

1000

1500

2000

2500

3000

Figure 4.2: The solution γ (in rad) for t f = 505s.

We have obtained the following solutions for h, γ,V and x. Please see Figures 4.1, 4.2, 4.3 and 4.4, respectively. We have set N = 3000 nodes and K = 100.

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120.05

120

119.95

119.9

119.85

119.8

119.75

119.7 0

500

1000

1500

2000

2500

3000

Figure 4.3: The solution V (in m/s) for t f = 505s. 104

9 8 7 6 5 4 3 2 1 0 0

500

1000

1500

2000

2500

3000

Figure 4.4: The solution x (in m) for t f = 505s.

4.3

Acknowledgements

The author is grateful to Professor Pedro Paglione of Technological Institute of Aeronautics, ITA, SP-Brazil, for his valuable suggestions and comments, which help me a lot to improve some important parts of this text, in particular on the part concerning the model in flight mechanics addressed.

CALCULUS OF VARIATIONS, CONVEX ANALYSIS AND RESTRICTED OPTIMIZATION

II

Chapter 5

Basic Topics on the Calculus of Variations

5.1

Banach spaces

We start by recalling the norm definition. Definition 5.1.1 Let V be a vectorial space. A norm in V is a function denoted by ∥ · ∥V : V → R+ = [0, +∞), for which the following properties hold: 1. ∥u∥V > 0, ∀u ∈ V such that u ̸= 0 and ∥u∥V = 0, if, and only if u = 0. 2. Triangular inequality, that is ∥u + v∥V ≤ ∥u∥V + ∥v∥V , ∀u, v ∈ V 3. ∥αu∥V = |α|∥u∥, ∀u ∈ V, α ∈ R. In such a case we say that the space V is a normed one. Definition 5.1.2 (Convergent sequence) Let V be a normed space and let {un } ⊂ V be a sequence. We say that {un } converges to u0 ∈ V , if for each ε > 0, there exists n0 ∈ N such that if n > n0 , then ∥un − u0 ∥V < ε.

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75

In such a case, we write lim un = u0 , in norm.

n→∞

Definition 5.1.3 (Cauchy sequence in norm) Let V be a normed space and let {un } ⊂ V be a sequence. We say that {un } is a Cauchy one, as for each ε > 0, there exists n0 ∈ N such that if m, n > n0 , then ∥un − um ∥V < ε. At this point we recall the definition of Banach space. Definition 5.1.4 (Banach space) A normed space V is said to be a Banach space as it is complete, that is, for each Cauchy sequence de Cauchy {un } ⊂ V there exists a u0 ∈ V such that ∥un − u0 ∥V → 0, as n → ∞. Example 5.1.5 Examples of Banach spaces: Consider V = C([a, b]), the space of continuous functions on [a, b]. We shall prove that such a space is a Banach one with the norm, ∥ f ∥V = max{| f (x)| : x ∈ [a, b]}. Exercise 5.1.6 Prove that ∥ f ∥V = max{| f (x)| : x ∈ [a, b]} is a norm for V = C([a, b]). Solution: 1. Clearly ∥ f ∥V ≥ 0, ∀ f ∈ V and ∥ f ∥V = 0 if, and only if f (x) = 0, ∀x ∈ [a, b], that is if, and only if f = 0. 2. Let f , g ∈ V . Thus, ∥ f + g∥V

= ≤ ≤ =

max{| f (x) + g(x)|, x ∈ [a, b]} max{| f (x)| + |g(x)|, x ∈ [a, b]} max{| f (x)|, x ∈ [a, b]} + max{|g(x)| x ∈ [a, b]} ∥ f ∥V + ∥g∥V . (5.1)

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3. Finally, let α ∈ R and f ∈ V. Hence, ∥α f ∥V

= = = =

max{|α f (x)|, x ∈ [a, b]} max{|α|| f (x)|, x ∈ [a, b]} |α| max{| f (x)|, x ∈ [a, b]} |α|∥ f ∥.

(5.2)

From this we may infer that ∥ · ∥V is a norm for V . The solution is complete. Theorem 5.1.7 V = C([a, b]) is a Banach space with the norm ∥ f ∥V = max{| f (x)| : x ∈ [a, b]}, ∀ f ∈ V. Proof 5.1 The proof that C([a, b]) is a vector space is left as an exercise. From the last exercise, ∥ · ∥V is a norm for V . Let { fn } ⊂ V be a Cauchy sequence. We shall prove that there exists f ∈ V such that ∥ fn − f ∥V → 0, as n → ∞. Let ε > 0. Thus, there exists n0 ∈ N such that if m, n > n0 , then ∥ fn − fm ∥V < ε. Hence max{| fn (x) − fm (x)| : x ∈ [a, b]} < ε, that is, | fn (x) − fm (x)| < ε, ∀x ∈ [a, b], m, n > n0 . Let x ∈ [a, b]. From (5.3), { fn (x)} is a real Cauchy sequence, therefore it is convergent. So, define f (x) = lim fn (x), ∀x ∈ [a, b]. n→∞

Also from (5.3), we have that lim | fn (x) − fm (x)| = | fn (x) − f (x)| ≤ ε, ∀n > n0 .

m→∞

From this we may infer that ∥ fn − f ∥V → 0, as n → ∞. We shall prove now that f is continuous on [a, b].

(5.3)

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77

From the exposed above, fn → f uniformly on [a, b] as n → ∞. Thus, there exists n1 ∈ N such that if n > n1 , then ε | fn (x) − f (x)| < , ∀x ∈ [a, b]. 3 Choose n2 > n1 . Let x ∈ [a, b]. From lim fn2 (y) = fn2 (x),

y→x

there exists δ > 0 such that if y ∈ [a, b] and |y − x| < δ , then ε | fn2 (y) − fn2 (x)| < . 3 Thus, if y ∈ [a, b] e |y − x| < δ , then | f (y) − f (x)| = | f (y) − fn2 (y) + fn2 (y) − fn2 (x) + fn2 (x) − f (x)| ≤ | f (y) − fn2 (y)| + | fn2 (y) − fn2 (x)| + | fn2 (x) − f (x)| ε ε ε + + < 3 3 3 = ε. (5.4) So, we may infer that f is continuous at x, ∀x ∈ [a, b], that is f ∈ V. The proof is complete. Exercise 5.1.8 Let V = C1 ([a, b]) be the space of functions f : [a, b] → R which the the first derivative is continuous on [a, b]. Define s function (in fact a functional) ∥ · ∥V : V → R+ by ∥ f ∥V = max{| f (x)| + | f ′ (x)| : x ∈ [a, b]}. 1. Prove that ∥ · ∥V is a norm. 2. Prove that V is a Banach space with such a norm. Solution: The proof of item 1 is left as an exercise. Now, we shall prove that V is complete. Let { fn } ⊂ V be a Cauchy sequence. Let ε > 0. Thus, there exists n0 ∈ N such that if m, n > n0 , then ∥ fn − fm ∥V < ε/2. Therefore, | fn (x) − fm (x)| + | fn′ (x) − fm′ (x)| < ε/2, ∀x ∈ [a, b], m, n > n0 .

(5.5)

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Let x ∈ [a, b]. hence, { fn (x)} are { fn′ (x)} real Cauchy sequences, and therefore, they are convergent. Denote f (x) = lim fn (x) n→∞

and g(x) = lim fn′ (x). n→∞

From this and (5.5), we obtain | fn (x) − f (x)| + | fn′ (x) − g(x)| =

lim | fn (x) − fm (x)| + | fn′ (x) − fm′ (x)|

m→∞

≤ ε/2, ∀x ∈ [a, b], n > n0 .

(5.6)

Similarly to the last example, we may obtain that f and g are continuous, therefore uniformly continuous on the compact set [a, b]. Thus, there exists δ > 0 such that if x, y ∈ [a, b] and |y − x| < δ , then |g(y) − g(x)| < ε/2.

(5.7)

Choose n1 > n0 . Let x ∈ (a, b). Hence, if 0 < |h| < δ , then from (5.6) and (5.7) we have fn1 (x + h) − fn1 (x) − g(x) h = | fn′ 1 (x + th) − g(x + th) + g(x + th) − g(x)| ≤ | fn′ 1 (x + th) − g(x + th)| + |g(x + th) − g(x)| < ε/2 + ε/2 = ε,

(5.8)

where from mean value theorem, t ∈ (0, 1) (it depends on h). Therefore, letting n1 → ∞, we get fn1 (x + h) − fn1 (x) − g(x) h f (x + h) − f (x) → − g(x) h ≤ ε, ∀0 < |h| < δ . (5.9) From this we may infer that f ′ (x) = lim

h→0

f (x + h) − f (x) = g(x), ∀x ∈ (a, b). h

The cases in which x = a or x = b may be dealt similarly with one-sided limits.

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From this and (5.6), we have ∥ fn − f ∥V → 0, as n → ∞ e f ∈ C1 ([a, b]). The solution is complete. Definition 5.1.9 (Functional) Let V be a Banach space. A functional F defined on V is a function whose the co-domain is R (F : V → R). Example 5.1.10 Let V = C([a, b]) and F : V → R where Z b

F(y) =

( sen3 x + y(x)2 ) dx, ∀y ∈ V.

a

Example 5.1.11 Let V = C1 ([a, b]) and let J : V → R where J(y) =

Z bq

1 + y′ (x)2 dx, ∀y ∈ C1 ([a, b]).

a

In our first frame-work we consider functionals defined as Z b

F(y) =

f (x, y(x), y′ (x)) dx,

a

where we shall assume f ∈ C([a, b] × R × R) and V = C1 ([a, b]). Thus, for F : D ⊂ V → R where Z b

F(y) =

f (x, y(x), y′ (x)) dx,

a

we assume V = C1 ([a, b]), and D = {y ∈ V : y(a) = A and y(b) = B}, where A, B ∈ R. Observe that if y ∈ D, then y + v ∈ D if, and only if, v ∈ V and v(a) = v(b) = 0. Indeed, in such a case y+v ∈V e y(a) + v(a) = y(a) = A,

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and y(b) + v(b) = y(b) = B. Thus, we define the space of admissible directions for F, denoted by Va , as, Va = {v ∈ V : v(a) = v(b) = 0}. Definition 5.1.12 (Global minimum) Let V be a Banach space and let F : D ⊂ V → R be a functional. We say that y0 ∈ D is a point of global minimum for F, if F(y0 ) ≤ F(y), ∀y ∈ D. Observe that denoting y = y0 + v where v ∈ Va , we have F(y0 ) ≤ F(y0 + v), ∀v ∈ Va . Example 5.1.13 Consider J : D ⊂ V → R where V = C1 ([a, b]), D = {y ∈ V : y(a) = 0 e y(b) = 1} and

Z b

J(y) =

(y′ (x))2 dx.

a

Thus, Va = {v ∈ V : v(a) = v(b) = 0}. Let y0 ∈ D be a candidate to global minimum for F and let v ∈ Va be admissible direction. Hence, we must have (5.10) J(y0 + v) − J(y0 ) ≥ 0, where J(y0 + v) − J(y0 ) =

Z b a

(y′0 (x) + v′ (x))2 dx −

Z b

= 2 a

≥ 2

Z b a

y′0 (x)v′ (x) dx +

Z b

Z b a

y′0 (x)2 dx

v′ (x)2 dx

a

y′0 (x)v′ (x) dx.

(5.11)

Observe that if y′0 (x) = c em [a, b], we have (5.10) satisfied, since in such a case, J(y0 + v) − J(y0 ) ≥ 2

Z b a

y′0 (x)v′ (x) dx

Z b

= 2c

v′ (x) dx

a

= 2c[v(x)]ba = 2c(v(b) − v(a)) = 0.

(5.12)

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Summarizing, if y′0 (x) = con [a, b], then J(y0 + v) ≥ J(y0 ), ∀v ∈ Va . Observe that in such a case, y0 (x) = cx + d, for some d ∈ R. However, from y(a) = 0, we get ca + d = 0. From y0 (b) = 1, we have cb + d = 1. Solving this last system in c and d we obtain, c=

1 , b−a

d=

−a . b−a

and

From this, we have,

x−a . b−a Observe that the graph of y0 corresponds to the straight line connecting the points (a, 0) and (b, 1). y0 (x) =

5.2

The Gˆateaux variation

Definition 5.2.1 Let V be a Banach space and let J : D ⊂ V → R be a functional. Let y ∈ D and v ∈ Va . We define the Gˆateaux variation of J at y in the direction v, denoted by δ J(y; v), by J(y + εv) − J(y) , δ J(y; v) = lim ε→0 ε if such a limit exists. Equivalently, δ J(y; v) =

∂ J(y + εv) |ε=0 . ∂ε

Example 5.2.2 Let V = C1 ([a, b]) and J : V → R where Z b

F(y) =

( sen3 x + y(x)2 ) dx.

a

Let y, v ∈ V . Let us calculate δ J(y; v).

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Observe that, J(y + εv) − J(y) ε Rb R ( sen3 x + (y(x) + εv(x))2 ) dx − ab ( sen3 x + y(x)2 ) dx = lim a ε→0 ε Rb 2 (2εy(x)v(x) + ε v(x)) dx = lim a ε→0 ε Z b Z b 2y(x)v(x) dx + ε = lim v(x)2 dx

δ J(y; v) =

lim

ε→0

ε→0

a

a

Z b

=

2y(x)v(x) dx.

(5.13)

a

Example 5.2.3 Let V = C1 ([a, b]) and let J : V → R where Z b q J(y) = ρ(x) 1 + y′ (x)2 dx, a

and where ρ : [a, b] → (0, +∞) is a fixed function. Let y, v ∈ V . Thus, δ J(y; v) = where

Z b

J(y + εv) =

∂ J(y + εv) |ε=0 , ∂ε

(5.14)

q ρ(x) 1 + (y′ (x) + εv′ (x))2 dx.

a

Hence, ∂ J(y + εv) |ε=0 ∂ε

= =(∗) =

Z

b

q ′ ′ 2 ρ(x) 1 + (y (x) + εv (x)) dx a q Z b ∂ ρ(x) 1 + (y′ (x) + εv′ (x))2 dx ∂ε a Z b ρ(x) 2(y′ (x) + εv′ (x))v′ (x) p dx. 2 a 1 + (y′ (x) + εv′ (x))2 ∂ ∂ε

(5.15)

(*): We shall prove this step is valid in the subsequent pages. From this we get, ∂ J(y + εv) |ε=0 ∂ε Z b ρ(x)y′ (x)v′ (x) p = dx. a 1 + y′ (x)2

δ J(y; v) =

The example is complete.

(5.16)

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Example 5.2.4 Let V = C1 ([a, b]) and f ∈ C1 ([a, b] × R × R). Thus f is a function of three variables, namely, f (x, y, z). Consider the functional F : V → R, defined by Z b

f (x, y(x), y′ (x)) dx.

F(y) = a

Let y, v ∈ V . Thus, δ F(y; v) =

∂ F(y + εv)|ε=0 . ∂ε

Observe que Z b

F(y + εv) =

f (x, y(x) + εv(x), y′ (x) + εv′ (x)) dx,

a

and therefore ∂ F(y + εv) = ∂ε =

∂ ∂ε

b

f (x, y(x) + εv(x), y′ (x) + εv′ (x)) dx

a

Z b ∂ a

=

Z

∂ε

f (x, y(x) + εv(x), y′ (x) + εv′ (x)) dx

Z b ∂ f (x, y(x) + εv(x), y′ (x) + εv′ (x))

v(x) ∂y ∂ f (x, y(x) + εv(x), y′ (x) + εv′ (x)) ′ v (x) dx. (5.17) + ∂z a

Thus ∂ F(y + εv) |ε=0 ∂ε Z b ∂ f (x, y(x), y′ (x)) ′ ∂ f (x, y(x), y′ (x)) v(x) + v (x) dx. (5.18) = ∂y ∂z a

δ F(y; v) =

5.3

Minimization of convex functionals

Definition 5.3.1 (Convex function) A function f : Rn → R is said to be convex if f (λ x + (1 − λ )y) ≤ λ f (x) + (1 − λ ) f (y), ∀x, y ∈ Rn , λ ∈ [0, 1]. Proposition 5.3.2 Let f : Rn → R be a convex and differentiable function. Under such hypotheses, f (y) − f (x) ≥ ⟨ f ′ (x), y − x⟩Rn , ∀x, y ∈ Rn , where ⟨·, ·⟩Rn : Rn × Rn → R denotes the usual inner product for Rn , that is, ⟨x, y⟩Rn = x1 y1 + · · · + xn yn , ∀x = (x1 , · · · , xn ), y = (y1 , · · · , yn ) ∈ Rn .

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Proof 5.2 Choose x, y ∈ Rn . From the hypotheses, f ((1 − λ )x + λ y) ≤ (1 − λ ) f (x) + λ f (y), ∀λ ∈ (0, 1). Thus, f (x + λ (y − x)) − f (x) ≤ f (y) − f (x), ∀λ ∈ (0, 1). λ Therefore, ⟨ f ′ (x), y − x⟩Rn

= ≤

f (x + λ (y − x)) − f (x) λ f (y) − f (x), ∀x, y ∈ Rn . lim

λ →0+

(5.19)

The proof is complete. Proposition 5.3.3 Let f : Rn → R be a differentiable function on Rn . Assume f (y) − f (x) ≥ ⟨ f ′ (x), y − x⟩Rn , ∀x, y ∈ Rn . Under such hypotheses, f is convex. Proof 5.3

Define f ∗ : Rn → R ∪ {+∞} by f ∗ (x∗ ) = sup {⟨x, x∗ ⟩Rn − f (x)}. x∈Rn

Such a function is said to the polar function for f . Let x ∈ Rn . From the hypotheses, ⟨ f ′ (x), x⟩Rn − f (x) ≥ ⟨ f ′ (x), y⟩Rn − f (y), ∀y ∈ Rn , that is, f ∗ ( f ′ (x)) =

sup {⟨ f ′ (x), y⟩Rn − f (y)}

y∈Rn ′

= ⟨ f (x), x⟩Rn − f (x).

(5.20)

On the other hand f ∗ (x∗ ) ≥ ⟨x, x∗ ⟩Rn − f (x), ∀x, x∗ ∈ Rn , and thus f (x) ≥ ⟨x, x∗ ⟩Rn − f ∗ (x∗ ), ∀x∗ ∈ Rn . Hence, f (x) ≥

sup {⟨x, x∗ ⟩Rn − f ∗ (x∗ )}

x∗ ∈Rn ′

≥ ⟨ f (x), x⟩Rn − f ∗ ( f ′ (x)).

(5.21)

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From this and (5.20), we obtain, f (x) =

sup {⟨x, x∗ ⟩Rn − f ∗ (x∗ )}

x∗ ∈Rn ′

= ⟨ f (x), x⟩Rn − f ∗ ( f ′ (x)).

(5.22)

Summarizing, f (x) = sup {⟨x, x∗ ⟩Rn − f ∗ (x∗ )}, ∀x ∈ Rn . x∗ ∈Rn

Rn

Choose x, y ∈ and λ ∈ [0, 1]. From the last equation we may write, f (λ x + (1 − λ )y) = =

sup {⟨λ x + (1 − λ )y, x∗ ⟩Rn − f ∗ (x∗ )}

x∗ ∈Rn

sup {⟨λ x + (1 − λ )y, x∗ ⟩Rn

x∗ ∈Rn

−λ f ∗ (x∗ ) − (1 − λ ) f ∗ (x∗ )} = sup {λ (⟨x, x∗ ⟩Rn − f ∗ (x∗ )) x∗ ∈Rn

+(1 − λ )(⟨y, x∗ ⟩Rn − f ∗ (x∗ ))} ≤ λ sup {⟨x, x∗ ⟩Rn − f ∗ (x∗ )} x∗ ∈Rn

+(1 − λ ) sup {⟨y, x∗ ⟩Rn − f ∗ (x∗ )} x∗ ∈Rn

= λ f (x) + (1 − λ ) f (y).

(5.23)

Since x, y ∈ Rn and λ ∈ [0, 1] are arbitrary, we may infer that f is convex. This completes the proof. Definition 5.3.4 (Convex functional) Let V be a Banach space and let J : D ⊂ V → R be a functional. We say that J is convex if J(y + v) − J(y) ≥ δ J(y; v), ∀v ∈ Va (y), where Va (y) = {v ∈ V : y + v ∈ D}. Theorem 5.3.5 Let V be a Banach space and let J : D ⊂ U be a convex functional. Thus, if y0 ∈ D is such that δ J(y0 ; v) = 0, ∀v ∈ Va (y0 ), then J(y0 ) ≤ J(y), ∀y ∈ D, that is, y0 minimizes J on D.

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Proof 5.4

Choose y ∈ D. Let v = y − y0 . Thus y = y0 + v ∈ D so that v ∈ Va (y0 ).

From the hypothesis, δ J(y0 ; v) = 0, and since J is convex, we obtain J(y) − J(y0 ) = J(y0 + v) − J(y0 ) ≥ δ J(y0 ; v) = 0, that is, J(y0 ) ≤ J(y), ∀y ∈ D. The proof is complete. Example 5.3.6 Let us see this example of convex functional. Let V = C1 ([a, b]) and let J : D ⊂ V → R be defined by Z b

J(y) =

(y′ (x))2 dx,

a

where D = {y ∈ V : y(a) = 1 e y(b) = 5}. We shall show that J is convex. Indeed, let y ∈ D e v ∈ Va where Va = {v ∈ V : v(a) = v(b) = 0}. Thus, J(y + v) − J(y) =

Z b

(y′ (x) + v′ (x))2 dx −

a

Z b

= Z b

y′ (x)2 dx

a

2y′ (x)v′ (x) dx +

a

≥

Z b

Z b

v′ (x)2 dx

a

2y′ (x)v′ (x) dx

a

= δ J(y; v). Therefore, J is convex.

(5.24)

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Sufficient conditions of optimality for the convex case

We start this section with a remark. Remark 5.4.1 Consider a function f : [a, b] × R × R → R where f ∈ C1 ([a, b] × R × R). Thus, for V = C1 ([a, b]), define F : V → R by Z b

F(y) =

f (x, y(x), y′ (x)) dx.

a

Let y, v ∈ V . We have already shown that Z b

δ F(y; v) = a

( fy (x, y(x), y′ (x))v(x) + fz (x, y(x), y′ (x))v′ (x)) dx.

Suppose f is convex in (y, z) for all x ∈ [a, b], which we denote by f (x, y, z) to be convex. From the last section, we have that f (x, y + v, y′ + v′ ) − f (x, y, y′ ) ≥ ⟨∇ f (x, y, y′ ), (v, v′ )⟩R2 = fy (x, y, y′ )v + fz (x, y, y′ )v′ , ∀x ∈ [a, b] (5.25) where we denote ∇ f (x, y, y′ ) = ( fy (x, y, y′ ), fz (x, y, y′ )). Therefore, F(y + v) − F(y) =

Z b

[ f (x, y + v, y′ + v′ ) − f (x, y, y′ )] dx

a

≥

Z b a

[ fy (x, y, y′ )v + fz (x, y, y′ )v′ ] dx

= δ J(y; v).

(5.26)

Thus, F is convex. Theorem 5.4.2 Let V = C1 ([a, b]). Let f ∈ C2 ([a, b] × R × R) where f (x, y, z) is convex. Define D = {y ∈ V : y(a) = a1 e y(b) = b1 }, where a1 , b1 ∈ R. Define also F : D → R by Z b

F(y) = a

f (x, y(x), y′ (x)) dx.

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Under such hypotheses, F is convex and if y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b], dx then y0 minimizes F on D, that is, F(y0 ) ≤ F(y), ∀y ∈ D. Proof 5.5

From the last remark, F is convex. Suppose now that y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b]. dx

Let v ∈ Va = {v ∈ V : v(a) = v(b) = 0}. Thus, Z b

( fy (x, y0 (x), y′0 (x))v(x) + fz (x, y0 (x), y′0 (x))v′ (x)) dx Z b d ′ ′ ′ ( fz (x, y0 (x), y0 (x))v(x)) + fz (x, y0 (x), y0 (x))v (x) dx = dx a Z b d ′ [ fz (x, y0 (x), y0 (x))v(x)] dx = dx a

δ F(y0 ; v) =

a

= [ fz (x, y0 (x), y′0 (x))v(x)]ba = fz (b, y0 (b), y′0 (b))v(b) − fz (a, y0 (a), y′0 (a))v(a) = 0, ∀v ∈ Va .

(5.27)

Since F is convex, from this and Theorem 5.3.5, we may infer that y0 minimizes J on D. Example 5.4.3 Let V = C1 ([a, b]) and D = {y ∈ V : y(0) = 0 and y(1) = 1}. Define F : D → R by Z 1

F(y) =

[y′ (x)2 + 5y(x)] dx, ∀y ∈ D.

0

Observe that

Z 1

F(y) =

f (x, y, y′ ) dx

0

where f (x, y, z) = z2 + 5y, that is, f (x, y, z) is convex.

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Thus, from the last theorem F is convex and if y0 ∈ D is such that d fz (x, y0 (x), y′0 (x)) = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b], dx then y0 minimizes F on D. Considering that fz (x, y, z) = 2z and fy (x, y, z) = 5, from this last equation we get, d (2y′ (x)) = 5, dx 0 that is, 5 y′′0 (x) = , ∀x ∈ [0, 1]. 2 Thus, 5 y′0 (x) = x + c, 2 and

5 y0 (x) = x2 + cx + d. 4 From this and y0 (0) = 0, we obtain d = 0. From this and y0 (1) = 1, we have 5 + c = 1, 4

de modo que c = −1/4 Therefore y0 (x) =

5x2 x − 4 4

minimizes F on D. The example is complete.

5.5

Natural conditions, problems with free extremals

We start this section with the following theorem. Theorem 5.5.1 Let V = C1 ([a, b]). Let f ∈ C2 ([a, b] × R × R) be such that f (x, y, z) is convex. Define D = {y ∈ V : y(a) = a1 }, where a1 ∈ R. Define also F : D → R by Z b

F(y) = a

f (x, y(x), y′ (x)) dx.

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Under such hypotheses,F is convex and if y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b] dx and fz (b, y0 (b), y′0 (b)) = 0 then y0 minimizes F on D, that is, F(y0 ) ≤ F(y), ∀y ∈ D. Proof 5.6 Since f (x, y, z) is convex from the last remark F is convex. Suppose now that y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b] dx and fz (b, y0 (b), y′0 (b)) = 0. Let v ∈ Va = {v ∈ V : v(a) = 0}. Thus, Z b

( fy (x, y0 (x), y′0 (x))v(x) + fz (x, y0 (x), y′0 (x))v′ (x)) dx Z b d ′ ′ ′ ( fz (x, y0 (x), y0 (x))v(x)) + fz (x, y0 (x), y0 (x))v (x) dx = dx a Z b d ′ [ fz (x, y0 (x), y0 (x))v(x)] dx = dx a

δ F(y0 ; v) =

a

= [ fz (x, y0 (x), y′0 (x))v(x)]ba = fz (b, y0 (b), y′0 (b))v(b) − fz (a, y0 (a), y′0 (a))v(a) = 0v(b) − fz (a, y0 (a), y′0 (b))0 = 0, ∀v ∈ Va .

(5.28)

Since F is convex, from this and Theorem 5.3.5, we may infer that y0 minimizes J on D. Remark 5.5.2 About this last theorem y(a) = a1 is said to be an essential boundary condition, whereas fz (b, y0 (b), y′0 (b)) = 0 is said to be a natural boundary condition. Theorem 5.5.3 Let V = C1 ([a, b]). Let f ∈ C2 ([a, b] × R × R) where f (x, y, z) is convex. Define D=V and F : D → R by

Z b

F(y) = a

f (x, y(x), y′ (x)) dx.

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Under such hypotheses, F is convex and if y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b], dx fz (a, y0 (a), y′0 (a)) = 0 e fz (b, y0 (b), y′0 (b)) = 0, then y0 minimizes F on D, that is, F(y0 ) ≤ F(y), ∀y ∈ D. Proof 5.7

From the last remark F is convex. Suppose that y0 ∈ D is such that d [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)), ∀x ∈ [a, b] dx

and fz (a, y0 (a), y′0 (a)) = fz (b, y0 (b), y′0 (b)) = 0. Let v ∈ D = V . Thus, Z b

( fy (x, y0 (x), y′0 (x))v(x) + fz (x, y0 (x), y′0 (x))v′ (x)) dx Z b d ( fz (x, y0 (x), y′0 (x))v(x)) + fz (x, y0 (x), y′0 (x))v′ (x) dx = dx a Z b d [ fz (x, y0 (x), y′0 (x))v(x)] dx = dx a

δ F(y0 ; v) =

a

= [ fz (x, y0 (x), y′0 (x))v(x)]ba = fz (b, y0 (b), y′0 (b))v(b) − fz (a, y0 (a), y′0 (a))v(a) = 0v(b) − 0v(a) = 0, ∀v ∈ D.

(5.29)

Since F is convex, from this and from Theorem 5.3.5, we may conclude that y0 minimizes J on D = V . The proof is complete. Remark 5.5.4 About this last theorem, the conditions fz (a, y0 (a), y′0 (a)) = fz (b, y0 (b), y′0 (b)) = 0 are said to be natural boundary conditions and the problem in question a free extremal one. Exercise 5.5.5 Show that F is convex and obtain its point of global minimum on D, D1 and D2 , where Z 2 ′ 2 y (x) dx, F(y) = x 1

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and where 1. D = {y ∈ C1 ([1, 2]) : y(1) = 0, y(1) = 3}, 2. D1 = {y ∈ C1 ([1, 2]) : y(2) = 3}. 3. D2 = C1 ([1, 2]). Solution: Observe that Z 2

F(y) =

f (x, y(x), y′ (x)) dx,

1

where f (x, y, z) = z2 /x, so that f (x, y, z) is convex. Therefore, F is convex. Let y, v ∈ V , thus, Z 2

δ F(y; v) = 1

[ fy (x, y, y′ )v + fz (x, y, y′ )v′ ] dx,

where fy (x, y, z) = 0 e fz (x, y, z) = 2z/x. Therefore, Z 2

δ F(y; v) =

2x−1 y′ (x)v′ (x) dx.

1

For D, from Theorem 5.7.1, sufficient conditions of optimality are given by, d dx [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)) em [1, 2], (5.30) y (1) = 0, 0 y0 (2) = 3. Thus, we must have d [2x−1 y′0 (x)] = 0, dx that is, 2x−1 y′0 (x) = c, so that y′0 (x) =

cx . 2

Therefore, y0 (x) =

cx2 + d. 4

Basic Topics on the Calculus of Variations

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93

On the other hand, we must have also y0 (1) =

c + d = 0, 4

and y0 (2) = c + d = 3. Thus, c = 4 and d = −1 so that y0 (x) = x2 − 1 minimizes F on D. For D1 , from Theorem 5.5.1, sufficient conditions of global optimality are given by d dx [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)) em [1, 2], (5.31) y (2) = 3, 0 fz (1, y0 (1), y′0 (1)) = 0. Thus, we must have y0 (x) =

cx2 + d. 4

On the other hand, we must also have y0 (2) = c + d = 3, and fz (1, y0 (1), y′0 (1)) = 2(1)−1 y′0 (1) = 0, that is, y′0 (1) = c/2 = 0, Therefore, c = 0 and d = 3 so that y0 (x) = 3 minimizes F on D1 . For D2 , from Theorem 5.5.3, sufficient conditions of global optimality are given by d dx [ fz (x, y0 (x), y′0 (x))] = fy (x, y0 (x), y′0 (x)) em [1, 2], (5.32) f (1, y0 (1), y′0 (1)) = 0 z fz (2, y0 (2), y′0 (2)) = 0. Thus, we have y0 (x) =

cx2 + d. 4

On the other hand we have also fz (1, y0 (1), y′0 (1)) = 2(1)−1 y′0 (1) = 0, fz (2, y0 (2), y′0 (2)) = 2(2)−1 y′0 (2) = 0, that is, y′0 (1) = y′0 (2) = 0, where y′0 (x) = cx/2. Thus c = 0, so that y0 (x) = d, ∀d ∈ R minimizes F on D2 .

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The Method of Lines and Duality Principles for Non-Convex Models

Exercise 5.5.6 Let V = C2 ([0, 1]) and J : D ⊂ V → R where J(y) =

EI 2

Z 1

y′′ (x)2 dx −

0

Z 1

P(x)y(x) dx, 0

represents the energy of a straight beam with rectangular cross section with inertial moment I. Here y(x) denotes the vertical displacement of the point x ∈ [0, 1] resulting from the action of distributed vertical load P(x) = αx, ∀x ∈ [0, 1], where E > 0 is the Young modulus and α > 0 is a real constant. And also D = {y ∈ V : y(0) = y(1) = 0}. Under such hypotheses, 1. prove that F is convex. 2. Prove that if y0 ∈ D is such that d4 EI dx 4 [y0 (x)] = P(x), ∀x ∈ [0, 1], y′′0 (0) = 0, ′′ y0 (1) = 0,

(5.33)

then y0 minimizes F on D. 3. Find the optimal solution y0 ∈ D. Solution: Let y ∈ D and v ∈ Va = {v ∈ V : v(0) = v(1) = 0}. We recall that F(y + εv) − F(y) ε→0 ε R R (EI/2) 01 [(y′′ + εv′′ )2 − (y′′ )2 ] dx − 01 (P(y + εv) − P) dx = lim ε→0 ε Z 1 Z εEI 1 ′′ 2 ′′ ′′ = lim (EIy v − Pv) dx + (v ) dx ε→0 2 0 0

δ J(y; v) =

lim

Z 1

= 0

(EIy′′ v′′ − Pv) dx.

(5.34)

Basic Topics on the Calculus of Variations

■

95

On the other hand, J(y + v) − J(v) = (EI/2)

Z 1

[(y′′ + v′′ )2 − (y′′ )2 ] dx −

0

Z 1

= Z 1

(P(y + v) − P) dx

0

(EIy′′ v′′ − Pv) dx +

0

≥

Z 1

EI 2

Z 1

(v′′ )2 dx

0

(EIy′′ v′′ − Pv) dx

0

= δ J(y; v).

(5.35)

Since y ∈ D and v ∈ Va are arbitrary, we may infer that J is convex. Assume that y0 ∈ D is such that d4 EI dx 4 [y0 (x)] = P(x), ∀x ∈ [0, 1], y′′0 (0) = 0, ′′ y0 (1) = 0,

(5.36)

Thus, Z 1

δ J(y; v) =

(EIy′′ v′′ − Pv) dx

0

Z 1

=

(EIy′′ v′′ − EIy(4) v) dx

0

Z 1

= 0

Z 1

=

′′′

(EIy′′ v′′ + EIy v′ ) dx − [EIy′′′ (x)v(x)]ba ′′′

(EIy′′ v′′ + EIy v′ ) dx

0

Z 1

= 0

′′

(EIy′′ v′′ − EIy v′′ ) dx + [EIy′′ (x)v′ (x)]ba

= 0

(5.37)

Summarizing δ J(y0 ; v) = 0, ∀v ∈ Va . Therefore, since J is convex, we may conclude that y0 minimizes J on D. To obtain the solution of the ODE is question, we shall denote y0 (x) = y p (x) + yh (x), where a particular solution y p is given by y p (x) = EI

αx5 120EI ,

where claerly

d4 [y p (x)] = P(x), ∀x ∈ [0, 1]. dx4

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The Method of Lines and Duality Principles for Non-Convex Models

The homogeneous associated equation EI

d4 [yh (x)] = 0, dx4

has the following general solution yh (x) = ax3 + bx2 + cx + d, and thus, y0 (x) = y p (x) + yh (x) =

αx5 + ax3 + bx2 + cx + d. 120EI

From y0 (0) = 0, we obtain d = 0. 5α x4 + 3ax2 + 2bx + c e y′′0 (x) = Observe that y′0 (x) = 120EI From this and y′′0 (0) = 0, we get b = 0. De y′′0 (1) = 0, obtemos, α 3 1 + 6a 1 = 0, 6EI e assim α a=− . 36EI From such results and from y0 (1) = 0, we obtain

α 3 6EI x + 6ax + 2b.

α α α + a 13 + c 1 = − + c = 0, 120EI 120EI 36EI that is, α c= EI

1 1 − 36 120

=

7α . 360EI

Finally, we have that y0 (x) =

αx3 7αx αx5 − + 120EI 36EI 360EI

minimizes J on D. The solution is complete.

5.6

The du Bois-Reymond lemma

Lemma 5.6.1 (du Bois-Reymond) Suppose h ∈ C([a, b]) and Z b a

h(x)v′ (x) dx = 0, ∀v ∈ Va ,

where Va = {v ∈ C1 ([a, b]) : v(a) = v(b) = 0}.

Basic Topics on the Calculus of Variations

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97

Under such hypotheses, there exists c ∈ R such that h(x) = c, ∀x ∈ [a, b]. Proof 5.8

Let

Rb a

c= Define

Z x

v(x) =

h(t) dt . b−a

(h(t) − c) dt.

a

Thus, v′ (x) = h(x) − c, ∀x ∈ [a, b], so that v ∈ C1 ([a, b]). Moreover, Z a

v(a) =

(h(t) − c) dt = 0,

a

and Z b

v(b) =

(h(t) − c) dt =

a

Z b

h(t) dt − c(b − a) = c(b − a) − c(b − a) = 0,

a

so that v ∈ Va . Observe that, from this and the hypotheses, 0 ≤

Z b

(h(t) − c)2 dt

a

Z b

=

(h(t) − c)(h(t) − c) dt

a

Z b

=

(h(t) − c)v′ (t) dt

a

Z b

=

h(t)v′ (t) dt − c

a

Z b

v′ (t) dt

a

= 0 − c(v(b) − v(a)) = 0. Thus, Z b

(h(t) − c)2 dt = 0.

a

Since h is continuous, we may infer that h(x) − c = 0, ∀x ∈ [a, b], that is, h(x) = c, ∀x ∈ [a, b]. The proof is complete.

(5.38)

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The Method of Lines and Duality Principles for Non-Convex Models

Theorem 5.6.2 Let g, h ∈ C([a, b]) and suppose Z b a

(g(x)v(x) + h(x)v′ (x)) dx = 0, ∀v ∈ Va ,

where Va = {v ∈ C1 ([a, b]) : v(a) = v(b) = 0}. Under such hypotheses, h ∈ C1 ([a, b]) e h′ (x) = g(x), ∀x ∈ [a, b]. Proof 5.9

Define

Z x

G(x) =

g(t) dt. a

Thus, G′ (x) = g(x), ∀x ∈ [a, b]. Let v ∈ Va . From the hypotheses, Z b

0 =

[g(x)v(x) + h(x)v′ (x)] dx

a

Z b

= a

Z b

= a

[−G(x)v′ (x) + h(x)v′ (x)] dx + [G(x)v(x)]ba [−G(x) + h(x)]v′ (x) dx, ∀v ∈ Va .

(5.39)

From this and from the du Bois - Reymond lemma, we may conclude that −G(x) + h(x) = c, ∀x ∈ [a, b], for some c ∈ R. Thus g(x) = G′ (x) = h′ (x), ∀x ∈ [a, b], so that g ∈ C1 ([a, b]). The proof is complete. Lemma 5.6.3 (Fundamental lemma of calculus of variation for one dimension) Let g ∈ C([a, b]) = V. Assume Z b

a

g(x)v(x) dx = 0, ∀v ∈ Va ,

where again, Va = {v ∈ C1 ([a, b]) : v(a) = v(b) = 0}. Under such hypotheses, g(x) = 0, ∀x ∈ [a, b].

Basic Topics on the Calculus of Variations

Proof 5.10

■

99

It suffices to apply the last theorem for h ≡ 0.

Exercise 5.6.4 Let h ∈ C([a, b]). Suppose Z b a

where

h(x)w(x) dx = 0, ∀w ∈ D0 ,

Z b D0 = w ∈ C([a, b]) : w(x) dx = 0 . a

Show that there exists c ∈ R such that h(x) = c, ∀x ∈ [a, b]. Solution Define, as above indicated, Va = {v ∈ C1 ([a, b]) : v(a) = v(b) = 0}. Let v ∈ Va . Let w ∈ C([a, b]) be such that w(x) = v′ (x), ∀x ∈ [a, b]. Observe that Z b

Z b

w(x) dx = a

From this

Rb a

a

v′ (x) dx = [v(x)]ba = v(b) − v(a) = 0.

h(x)w(x) dx = 0, and thus Z b

h(x)v′ (x) dx = 0.

a

Since v ∈ Va is arbitrary, from this and the du Bois-Reymond lemma, there exists c ∈ R such that h(x) = c, ∀x ∈ [a, b]. The solution is complete.

5.7

Calculus of variations, the case of scalar functions on Rn

Let Ω ⊂ Rn be an open, bounded, connected with a regular boundary ∂ Ω = S (Lipschitzian) (which we define as Ω to be of class Cˆ 1 ). Let V = C1 (Ω) and let F : D ⊂ V → R, be such that Z

F(y) = Ω

f (x, y(x), ∇y(x)) dx, ∀y ∈ V,

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The Method of Lines and Duality Principles for Non-Convex Models

where we denote dx = dx1 · · · dxn . Assume f : Ω × R × Rn → R is of C2 class. Suppose also f (x, y, z) is convex in (y, z), ∀x ∈ Ω, which we denote by f (x, y, z) to be convex. Observe that for y ∈ D e v ∈ Va , where D = {y ∈ V : y = y1 on ∂ Ω}, and Va = {v ∈ V : v = 0 on ∂ Ω}, where y1 ∈ C1 (Ω), we have that δ F(y; v) = where

∂ F(y + εv)|ε=0 , ∂ε

Z

F(y + εv) =

f (x, y + εv, ∇y + ε∇v) dx. Ω

Therefore, ∂ F(y + εv) = ∂ε

Z Ω

∂ ( f (x, y + εv, ∇y + ε∇v) dx ∂ε n

Z

= Ω

[ fy (x, y + εv, ∇y + ε∇v)v + ∑ fzi (x, y + εv, ∇y + ε∇v)vxi ] dx. (5.40) i=1

Thus, δ F(y; v) =

∂ F(y + εv)|ε=0 ∂ε n

Z

= Ω

[ fy (x, y, ∇y)v + ∑ fzi (x, y, ∇y)vxi ] dx.

(5.41)

i=1

On the other hand, since f (x, y, z) is convex, we have that F(y + v) − F(y) =

Z

[ f (x, y + v, ∇y + ∇v) − f (x, y, ∇y)] dx

Ω

≥ ⟨∇ f (x, y, ∇y), (v, ∇v)⟩Rn+1 n

Z

= Ω

[ fy (x, y, ∇y)v + ∑ fzi (x, y, ∇y)vxi ] dx i=1

= δ F(y; v). Since y ∈ D and v ∈ Va are arbitrary, w e may infer that F is convex. Here we denote, ∇ f (x, y, ∇y) = ( fy (x, y, ∇y), fz1 (x, y, ∇y), · · · , fzn (x, y, ∇y)).

(5.42)

Basic Topics on the Calculus of Variations

101

■

Theorem 5.7.1 Let Ω ⊂ Rn be a set of Cˆ 1 class and let V = C1 (Ω). Let f ∈ C2 (Ω × R × R) where f (x, y, z) is convex. Define D = {y ∈ V : y = y1 em ∂ Ω}, where y1 ∈ C1 (Ω) Define also F : D → R by Z

F(y) =

f (x, y(x), ∇y(x)) dx. Ω

From such hypotheses, F is convex and if y0 ∈ D is such that n

d

∑ dxi [ fzi (x, y0 (x), ∇y0 (x))] = fy (x, y0 (x), ∇y0 (x)), ∀x ∈ Ω,

i=1

then y0 minimizes F on D, that is, F(y0 ) ≤ F(y), ∀y ∈ D. Proof 5.11 that

From the last remark, F is convex. Suppose now that y0 ∈ D is such n

d

∑ dxi [ fzi (x, y0 (x), ∇y0 (x))] = fy (x, y0 (x), ∇y0 (x)), ∀x ∈ Ω,

i=1

Let v ∈ Va = {v ∈ V : v = 0 on ∂ Ω}. Thus, n

Z

δ F(y0 ; v)

= Ω

( fy (x, y0 (x), ∇y0 (x))v(x) + ∑ fzi (x, y0 (x), ∇y0 (x))vxi (x)) dx i=1

n d ∑ dxi ( fzi (x, y0 (x), ∇y0 (x)))v(x) + ∑ fzi (x, y0 (x), ∇y0 (x))vxi (x) i=1 i=1 ! n

Z

= Ω

− ∑ fzi (x, y0 (x), ∇y0 (x))vxi (x) + ∑ fzi (x, y0 (x), ∇y0 (x))vxi (x)

Ω

Z

+

dx

n

n

Z

=

!

i=1 n

dx

i=1

∑ fz (x, y0 (x), ∇y0 (x)) ni v(x) dS i

∂ Ω i=1

=

0, ∀v ∈ Va ,

(5.43)

where n = (n1 , · · · , nn ) denotes the outward normal field to ∂ Ω = S. Since F is convex, from this and Theorem 5.3.5, we have that y0 minimizes F on D.

5.8

The second Gˆateaux variation

Definition 5.8.1 Let V be a Banach space. Let F : D ⊂ V → R be a functional such that δ F(y; v) exists on Br (y0 ) for y0 ∈ D, r > 0 and for all v ∈ Va .

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The Method of Lines and Duality Principles for Non-Convex Models

Let y ∈ Br (y0 ) e v, w ∈ Va . We define the second Gˆateaux variation of F at the point y in the directions v e w, denoted by δ 2 F(y; v, w), as δ F(y + εw; v) − δ F(y; v) , ε→0 ε

δ 2 F(y; v, w) = lim if such a limit exists.

Remark 5.8.2 Observe that from this last definition, if the limits in question exist, we have ∂ F(y + εv)|ε=0 , δ F(y; v) = ∂ε and ∂2 δ 2 F(y; v, v) = 2 F(y + εv)|ε=0 , ∀v ∈ Va . ∂ε 1 Thus, for example, for V = C (Ω) where Ω ⊂ Rn is of Cˆ 1 class and F : V → R is given by Z F(y) =

f (x, y, ∇y) dx Ω

and where f ∈ C2 (Ω × R × Rn ), para y, v ∈ V , we have δ 2 F(y; v, v) =

∂2 F(y + εv)|ε=0 , ∂ ε2

where ∂2 F(y + εv) ∂ ε2

=

∂2 ∂ ε2

f (x, y + εv, ∇y + ε∇v) dx

Ω

∂2

Z

=

Z

[ f (x, y + εv, ∇y + ε∇v)] dx ∂ ε2 " Z n fyy (x, y + εv, ∇y + ε∇v)v2 + ∑ 2 fyzi (x, y + εv, ∇y + ε∇v)vvxi Ω

=

Ω

i=1

n

#

n

+ ∑ ∑ fzi z j (x, y + εv, ∇y + ε∇v)vxi vx j

dx

(5.44)

i=1 j=1

so that δ 2 F(y; v, v) =

∂2 F(y + εv)|ε=0 2 ∂ ε"

Z

n

fyy (x, y, ∇y)v2 + ∑ 2 fyzi (x, y, ∇y)vvxi

= Ω

i=1

n

n

+ ∑ ∑ fzi z j (x, y, ∇y)vxi vx j i=1 j=1

# dx.

(5.45)

Basic Topics on the Calculus of Variations

5.9

■

103

First order necessary conditions for a local minimum

Definition 5.9.1 Let V be a Banach space. Let F : D ⊂ V → R be a functional. We say that y0 ∈ D is a point of local minimum for F on D, if there exists δ > 0 such that F(y) ≥ F(y0 ), ∀y ∈ Bδ (y0 ) ∩ D. Theorem 5.9.2 [First order necessary condition] Let V be a Banach space. Let F : D ⊂ V → R be a functional. Suppose that y0 ∈ D is a point of local minimum for F on D. Let v ∈ Va and assume δ F(y0 ; v) to exist. Under such hypotheses, δ F(y0 ; v) = 0. Proof 5.12 Define φ (ε) = F(y0 + εv), which from the existence of δ F(y0 ; v) is well defined for all ε sufficiently small. Also from the hypotheses, ε = 0 is a point of local minimum for the differentiable at 0 function φ . Thus, from the standard condition for one variable calculus, we have φ ′ (0) = 0, that is, φ ′ (0) = δ F(y0 ; v) = 0. The proof is complete. Theorem 5.9.3 (Second order sufficient condition) Let V be a Banach space. Let F : D ⊂ V → R be a functional. Suppose y0 ∈ D is such that δ F(y0 ; v) = 0 for all v ∈ Va and there exists δ > 0 such that δ 2 F(y; v, v) ≥ 0, ∀y ∈ Bδ (y0 ) and v ∈ Va . Under such hypotheses y0 ∈ D is a point of local minimum for F, that is F(y) ≥ F(y0 ), ∀y ∈ Br (y0 ) ∩ D. Proof 5.13 Let y ∈ Bδ (y0 ) ∩ D. Define v = y − y0 ∈ Va . Define also φ : [0, 1] → R by φ (ε) = F(y0 + εv). From the Taylor Theorem for one variable, there exists t0 ∈ (0, 1) such that φ (1) = φ (0) +

1 φ ′ (0) (1 − 0) + φ ′′ (t0 )(1 − 0)2 , 1! 2!

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The Method of Lines and Duality Principles for Non-Convex Models

That is, F(y) =

F(y0 + v)

1 = F(y0 ) + δ F(y0 ; v) + δ 2 F(y0 + t0 v; v, v) 2 1 = F(y0 ) + δ 2 F(y0 + t0 v; v, v) 2 ≥ F(y0 ), ∀y ∈ Bδ (y0 ) ∩ D.

(5.46)

The proof is complete.

5.10

Continuous functionals

Definition 5.10.1 Let V be a Banach space. Let F : D ⊂ V → R be a functional and let y0 ∈ D. We say that F is continuous on y0 ∈ D, if for each ε > 0 there exists δ > 0 such that if y ∈ D e ∥y − y0 ∥V < δ , then |F(y) − F(y0 )| < ε. Example 5.10.2 Let V = C1 ([a, b]) and f ∈ C([a, b] × R × R). Consider F : V → R where Z b

F(y) =

f (x, y(x), y′ (x)) dx,

a

and ∥y∥V = max{|y(x)| + |y′ (x)| : x ∈ [a, b]}. Let y0 ∈ V. We shall prove that F is continuous at y0 . Let y ∈ V be such that ∥y − y0 ∥V < 1. Thus, ∥y∥V − ∥y0 ∥V ≤ ∥y − y0 ∥V < 1, that is, ∥y∥V < 1 + ∥y0 ∥V ≡ α. Observe that f is uniformly continuous on the compact set [a, b] × [−α, α] × [−α, α] ≡ A. Let ε > 0. Therefore, there exists δ0 > 0 such that if (x, y1 , z1 ) e (x, y2 , z2 ) ∈ A e |y1 − y2 | + |z1 − z2 | < δ0 , then | f (x, y1 , z1 ) − f (x, y2 , z2 )|

j0 , then Z Z [Gn (x) − G(x)] dx ≤ |Gn j (x) − G(x)| dx j Ω

Ω

≤

Z

cn j dx Ω

= cn j m(Ω) < ε. Thus,

Z

Z

lim

j→∞ Ω

(5.51)

Gn j (x) dx =

G(x) dx. Ω

Suppose now,to obtain contradiction, that we do not have Z

lim ε→0 Ω

Z

Gε (x) dx =

G(x) dx, Ω

where Gε (x) =

f (x, y(x) + εv(x), ∇y(x) + ε∇v(x)) − f (x, y(x), ∇y(x)) , ε

∀ε ∈ R such that ε ̸= 0. Hence, there exists ε0 > 0 such that for each n ∈ N there exists ε˜n ∈ R such that 1 0 < |ε˜n | < , n and

Z Z G˜ n (x) dx − G(x) dx ≥ ε0 , Ω Ω

where f (x, y(x) + ε˜n v(x), ∇y(x) + ε˜n ∇v(x)) − f (x, y(x), ∇y(x)) , G˜ n (x) = ε˜n ∀n ∈ N, x ∈ Ω.

(5.52)

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The Method of Lines and Duality Principles for Non-Convex Models

However, as above indicated, we may obtain a subsequence {ε˜n j } of {ε˜n } such that Z Z lim G˜ n j (x) dx = G(x) dx, j→∞ Ω

Ω

which contradicts (5.52). Therefore, necessarily we have that Z

lim ε→0 Ω

Z

Gε (x) dx =

G(x) dx, Ω

that is, F(y + εv) − F(y) ε→0 ε Z

δ F(y; v) = lim = lim

ε→0 Ω

Gε (x) dx

Z

=

G(x) dx Ω n

Z

= Ω

!

fy (x, y(x), ∇y(x))v(x) + ∑ fzi (x, y(x), ∇y(x))vxi (x)

The proof is complete.

i=1

dx. (5.53)

Chapter 6

More Topics on the Calculus of Variations

6.1

Introductory remarks

We recall that a functional is a function whose the co-domain is the real set. We denote such functionals by F : U → R, where U is a Banach space. In our work format, we consider the special cases 1. F(u) =

R

2. F(u) =

R

Ω Ω

f (x, u, ∇u) dx, where Ω ⊂ Rn is an open, bounded, connected set. f (x, u, ∇u, D2 u) dx, here Du = ∇u =

and D2 u = {D2 ui } =

∂ ui ∂xj

∂ 2 ui ∂ xk ∂ xl

,

for i ∈ {1, ..., N} and j, k, l ∈ {1, ..., n}. Also, f : Ω × RN × RN×n → R is denoted by f (x, s, ξ ) and we assume 1.

∂ f (x, s, ξ ) ∂s

110

The Method of Lines and Duality Principles for Non-Convex Models

■

and 2.

∂ f (x, s, ξ ) ∂ξ are continuous ∀(x, s, ξ ) ∈ Ω × RN × RN×n .

Remark 6.1.1 We also recall that the notation ∇u = Du may be used. Now we define our general problem, namely problem P where Problem P : minimize F(u) on U, that is, to find u0 ∈ U such that F(u0 ) = min{F(u)}. u∈U

At this point, we introduce some essential definitions. Theorem 6.1.2 Consider the hypotheses stated at Section 6.1 on F : U → R. Sup¯ RN ) and additionally assume that pose F attains a local minimum at u ∈ C2 (Ω; N N×n 2 ). Then the necessary conditions for a local minimum for F are f ∈ C (Ω, R , R given by the Euler-Lagrange equations: ∂ f (x, u, ∇u) ∂ f (x, u, ∇u) − div = θ , in Ω. ∂s ∂ξ Proof 6.1 Observe that the standard first order necessary condition stands for δ F(u, ϕ) = 0, ∀ϕ ∈ V . From above this implies, after integration by parts Z Ω

∂ f (x, u, ∇u) ∂ f (x, u, ∇u) − div · ϕ dx = 0, ∂s ∂ξ ∀ϕ ∈ Cc∞ (Ω, RN ).

The result then follows from the fundamental lemma of calculus of variations.

6.2

The Gˆateaux variation, a more general case

Theorem 6.2.1 Consider the functional F : U → R, where U = {u ∈ W 1,2 (Ω, RN ) | u = u0 in ∂ Ω}. Suppose Z

F(u) =

f (x, u, ∇u) dx, Ω

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where f : Ω × RN × RN×n is such that, for each K > 0 there exists K1 > 0 such that | f (x, s1 , ξ1 ) − f (x, s2 , ξ2 )| < K1 (|s1 − s2 | + |ξ1 − ξ2 |) ∀s1 , s2 ∈ RN , ξ1 , ξ2 ∈ RN×n , such that |s1 | < K, |s2 | < K, |ξ1 | < K, |ξ2 | < K. Also assume the hypotheses of Section 6.1 except for the continuity of derivatives of f . Under such assumptions, for each u ∈ C1 (Ω; RN ) and ϕ ∈ Cc∞ (Ω; RN ), we have Z ∂ f (x, u, ∇u) ∂ f (x, u, ∇u) ·ϕ + · ∇ϕ dx. δ F(u, ϕ) = ∂s ∂ξ Ω Proof 6.2

First we recall that δ F(u, ϕ) = lim ε→0

F(u + εϕ) − F(u) . ε

Observe that lim ε→0

f (x, u + εϕ, ∇u + ε∇ϕ) − f (x, u, ∇u) ε ∂ f (x, u, ∇u) ∂ f (x, u, ∇u) = ·ϕ + · ∇ϕ, a.e in Ω. ∂s ∂ξ

Define G(x, u, ϕ, ε) = and

f (x, u + εϕ, ∇u + ε∇ϕ) − f (x, u, ∇u) , ε

˜ u, ϕ) = ∂ f (x, u, ∇u) · ϕ + ∂ f (x, u, ∇u) · ∇ϕ. G(x, ∂s ∂ξ

Thus we have ˜ u, ϕ), a.e in Ω. lim G(x, u, ϕ, ε) = G(x, ε→0

Now will show that Z

Z

˜ u, ϕ) dx. G(x,

G(x, u, ϕ, ε) dx =

lim ε→0 Ω

Ω

It suffices to show that (we do not provide details here) Z

Z

lim

n→∞ Ω

˜ u, ϕ) dx. G(x,

G(x, u, ϕ, 1/n) dx = Ω

Observe that, for an appropriate K > 0, we have |G(x, u, ϕ, 1/n)| ≤ K(|ϕ| + |∇ϕ|), a.e. in Ω.

(6.1)

By the Lebesgue dominated convergence theorem, we obtain Z

lim

Z

n→+∞ Ω

G(x, u, ϕ, 1/(n)) dx =

˜ u, ϕ) dx, G(x,

Ω

that is, Z

δ F(u, ϕ) = Ω

∂ f (x, u, ∇u) ∂ f (x, u, ∇u) ·ϕ + · ∇ϕ ∂s ∂ξ

dx.

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The Method of Lines and Duality Principles for Non-Convex Models

Fr´echet differentiability

In this section, we introduce a very important definition namely, Fr´echet differentiability. Definition 6.3.1 Let U,Y be Banach spaces and consider a transformation T : U → Y. We say that T is Fr´echet differentiable at u ∈ U if there exists a bounded linear transformation T ′ (u) : U → Y such that lim

v→θ

∥T (u + v) − T (u) − T ′ (u)(v)∥Y = 0, v ̸= θ . ∥v∥U

In such a case T ′ (u) is called the Fr´echet derivative of T at u ∈ U.

6.4

The Legendre-Hadamard condition

¯ RN ) is such that Theorem 6.4.1 If u ∈ C1 (Ω; δ 2 F(u, ϕ) ≥ 0, ∀ϕ ∈ Cc∞ (Ω, RN ), then fξαi ξ k (x, u(x), ∇u(x))ρ i ρ k ηα ηβ ≥ 0, ∀x ∈ Ω, ρ ∈ RN , η ∈ Rn . β

Such a condition is known as the Legendre-Hadamard condition. Proof 6.3

Suppose δ 2 F(u, ϕ) ≥ 0, ∀ϕ ∈ Cc∞ (Ω; RN ).

We denote δ 2 F(u, ϕ) by 2

Z

a(x)Dϕ(x) · Dϕ(x) dx

δ F(u, ϕ) = Ω

Z

b(x)ϕ(x) · Dϕ(x) dx +

+ Ω

Z

c(x)ϕ(x) · ϕ(x) dx,

(6.2)

Ω

where a(x) = fξ ξ (x, u(x), Du(x)), b(x) = 2 fsξ (x, u(x), Du(x)), and c(x) = fss (x, u(x), Du(x)). N Now consider v ∈ Cc∞ (B1 (0), R ). Thus given x0 ∈ Ω for λ sufficiently small we 0 is an admissible direction. Now we introduce the new have that ϕ(x) = λ v x−x λ

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coordinates y = (y1 , ..., yn ) by setting y = λ −1 (x − x0 ) and multiply (6.2) by λ −n to obtain Z B1 (0)

{a(x0 + λ y)Dv(y) · Dv(y) + 2λ b(x0 + λ y)v(y) · Dv(y) + λ 2 c(x0 + λ y)v(y) · v(y)} dy > 0, β

αβ

where a = {ai j }, b = {b jk } and c = {c jk }. Since a, b and c are continuous, we have a(x0 + λ y)Dv(y) · Dv(y) → a(x0 )Dv(y) · Dv(y), λ b(x0 + λ y)v(y) · Dv(y) → 0, and λ 2 c(x0 + λ y)v(y) · v(y) → 0, ¯ as λ → 0. Thus this limit give us uniformly on Ω Z B1 (0)

αβ f˜jk Dα v j Dβ vk dx ≥ 0, ∀v ∈ Cc∞ (B1 (0); RN ),

(6.3)

where αβ αβ f˜jk = a jk (x0 ) = fξαi ξ k (x0 , u(x0 ), ∇u(x0 )). β

Now define v = (v1 , ..., vN ) where v j = ρ j cos((η · y)t)ζ (y) ρ = (ρ 1 , ..., ρ N ) ∈ RN and η = (η1 , ..., ηn ) ∈ Rn and ζ ∈ Cc∞ (B1 (0)). From (6.3) we obtain Z αβ 0 ≤ f˜jk ρ j ρ k (ηα t(−sin((η · y)t)ζ + cos((η · y)t)Dα ζ ) B1 (0) · ηβ t(−sin((η · y)t)ζ + cos((η · y)t)Dβ ζ dy

(6.4)

By analogy for v j = ρ j sin((η · y)t)ζ (y) we obtain 0 ≤

αβ f˜jk ρ j ρ k

Z

(ηα t(cos((η · y)t)ζ + sin((η · y)t)Dα ζ ) · ηβ t(cos((η · y)t)ζ + sin((η · y)t)Dβ ζ dy B1 (0)

(6.5)

Summing up these last two equations, dividing the result by t 2 and letting t → +∞ we obtain Z αβ 0 ≤ f˜jk ρ j ρ k ηα ηβ ζ 2 dy, B1 (0)

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for all ζ ∈ Cc∞ (B1 (0)), which implies αβ 0 ≤ f˜jk ρ j ρ k ηα ηβ .

The proof is complete.

6.5

The Weierstrass condition for n = 1

Here we present the Weierstrass condition for the special case N ≥ 1 and n = 1. We start with a definition. ˆ Definition 6.5.1 We say that u ∈ C([a, b]; RN ) if u : [a, b] → RN is continuous in [a, b], and Du is continuous except on a finite set of points in [a, b]. ¯ × RN × RN → R be such that Theorem 6.5.2 (Weierstrass) Let Ω = (a, b) and f : Ω N ¯ fs (x, s, ξ ) and fξ (x, s, ξ ) are continuous on Ω × R × RN . Define F : U → R by Z b

F(u) =

f (x, u(x), u′ (x)) dx,

a

where U = {u ∈ Cˆ 1 ([a, b]; RN ) | u(a) = α, u(b) = β }. Suppose u ∈ U minimizes locally F on U, that is, suppose that there exists ε0 > 0 such that F(u) ≤ F(v), ∀v ∈ U, such that ∥u − v∥∞ < ε0 . Under such hypotheses, we have E(x, u(x), u′ (x+), w) ≥ 0, ∀x ∈ [a, b], w ∈ RN , and E(x, u(x), u′ (x−), w) ≥ 0, ∀x ∈ [a, b], w ∈ RN , where u′ (x+) = lim u′ (x + h), h→0+

′

u (x−) = lim u′ (x + h), h→0−

and, E(x, s, ξ , w) = f (x, s, w) − f (x, s, ξ ) − fξ (x, s, ξ )(w − ξ ). Remark 6.5.3 The function E is known as the Weierstrass Excess Function.

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Proof 6.4 Fix x0 ∈ (a, b) and w ∈ RN . Choose 0 < ε < 1 and h > 0 such that u + v ∈ U and ∥v∥∞ < ε0 where v(x) is given by (x − x0 )w, ε˜ (h − x + x0 )w, v(x) = 0, where ε˜ =

if 0 ≤ x − x0 ≤ εh, if εh ≤ x − x0 ≤ h, otherwise,

ε . 1−ε

From F(u + v) − F(u) ≥ 0 we obtain Z x0 +h

f (x, u(x) + v(x), u′ (x) + v′ (x)) dx

x0

−

Z x0 +h

f (x, u(x), u′ (x)) dx ≥ 0. (6.6)

x0

Define x˜ =

x − x0 , h

so that d x˜ =

dx . h

From (6.6) we obtain Z 1

h 0

f (x0 + xh, ˜ u(x0 + xh) ˜ + v(x0 + xh), ˜ u′ (x0 + xh) ˜ + v′ (x0 + xh) ˜ d x˜ −h

Z 1 0

f (x0 + xh, ˜ u(x0 + xh), ˜ u′ (x0 + xh)) ˜ d x˜ ≥ 0.

(6.7)

where the derivatives are related to x. Therefore Z ε 0

−

f (x0 + xh, ˜ u(x0 + xh) ˜ + v(x0 + xh), ˜ u′ (x0 + xh) ˜ + w) d x˜

Z ε 0

Z 1

+

f (x0 + xh, ˜ u(x0 + xh), ˜ u′ (x0 + xh)) ˜ d x˜ f (x0 + xh, ˜ u(x0 + xh) ˜ + v(x0 + xh), ˜ u′ (x0 + xh) ˜ − ε˜ w) d x˜

ε

−

Z 1

f (x0 + xh, ˜ u(x0 + xh), ˜ u′ (x0 + xh)) ˜ d x˜

ε

≥ 0.

(6.8)

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Letting h → 0 we obtain ε( f (x0 , u(x0 ), u′ (x0 +) + w) − f (x0 , u(x0 ), u′ (x0 +)) +(1 − ε)( f (x0 , u(x0 ), u′ (x0 +) − ε˜ w) − f (x0 , u(x0 ), u′ (x0 +))) ≥ 0. Hence, by the mean value theorem we get ε( f (x0 , u(x0 ), u′ (x0 +) + w) − f (x0 , u(x0 ), u′ (x0 +)) −(1 − ε)ε˜ ( fξ (x0 , u(x0 ), u′ (x0 +) + ρ(ε˜ )w)) · w ≥ 0.

(6.9)

Dividing by ε and letting ε → 0, so that ε˜ → 0 and ρ(ε˜ ) → 0 we finally obtain f (x0 , u(x0 ), u′ (x0 +) + w) − f (x0 , u(x0 ), u′ (x0 +)) − fξ (x0 , u( x0 ), u′ (x0 +)) · w ≥ 0. Similarly we may get f (x0 , u(x0 ), u′ (x0 −) + w) − f (x0 , u(x0 ), u′ (x0 −)) − fξ (x0 , u( x0 ), u′ (x0 −)) · w ≥ 0. Since x0 ∈ [a, b] and w ∈ RN are arbitrary, the proof is complete.

6.6

The Weierstrass condition, the general case

In this section, we present a proof for the Weierstrass necessary condition for N ≥ 1, n ≥ 1. Such a result may be found in similar form in [43]. Theorem 6.1 Assume u ∈ C1 (Ω; RN ) is a point of strong minimum for a Fr´echet differentiable functional F : U → R that is, in particular, there exists ε > 0 such that F(u + ϕ) ≥ F(u), for all ϕ ∈ Cc∞ (Ω; Rn ) such that ∥ϕ∥∞ < ε. Here

Z

F(u) =

f (x, u, Du) dx, Ω

where we recall to have denoted Du = ∇u =

∂ ui ∂xj

.

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Under such hypotheses, for all x ∈ Ω and each rank-one matrix η = {ρi β α } = {ρ ⊗ β }, we have that E(x, u(x), Du(x), Du(x) + ρ ⊗ β ) ≥ 0, where

=

Proof 6.5

E(x, u(x), Du(x), Du(x) + ρ ⊗ β ) f (x, u(x), Du(x) + ρ ⊗ β ) − f (x, u(x), Du(x)) −ρ i βα fξαi (x, u(x), Du(x)).

(6.10)

Since u is a point of local minimum for F, we have that δ F(u; ϕ) = 0, ∀ϕ ∈ Cc∞ (Ω; RN ),

that is

Z

(ϕ · fs (x, u(x), Du(x)) + Dϕ · fξ (x, u(x), Du(x)) dx = 0,

Ω

and hence, Z

( f (x, u(x), Du(x) + Dϕ(x)) − f (x, u(x), Du(x)) dx

Ω

−

Z

(ϕ(x) · fs (x, u(x), Du(x)) − Dϕ(x) · fξ (x, u(x), Du(x)) dx

Ω

≥ 0,

(6.11)

∀ϕ ∈ V , where V = {ϕ ∈ Cc∞ (Ω; RN ) : ∥ϕ∥∞ < ε}. Choose a unite vector e ∈ Rn and write x = (x · e)e + x, where x · e = 0. Denote De v = Dv · e, and let ρ = (ρ1 , ...., ρN ) ∈ RN . Also, let x0 be any point of Ω. Without loss of generality assume x0 = 0. Choose λ0 ∈ (0, 1) such that Cλ0 ⊂ Ω, where, Cλ0 = {x ∈ Rn : |x · e| ≤ λ0 and ∥x∥ ≤ λ0 }. Let λ ∈ (0, λ0 ) and φ ∈ Cc ((−1, 1); R) and choose a sequence φk ∈ Cc∞ ((−λ 2 , λ ); R)

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which converges uniformly to the Lipschitz function φλ given by if − λ 2 ≤ t ≤ 0, t + λ 2, λ (λ − t), if 0 < t < λ φλ = 0, otherwise

(6.12)

and such that φk′ converges uniformly to φλ′ on each compact subset of Aλ = {t : −λ 2 < t < λ , t ̸= 0}. We emphasize the choice of {φk } may be such that for some K > 0 we have ∥φ ∥∞ < K, ∥φk ∥∞ < K and ∥φk′ ∥∞ < K, ∀k ∈ N. Observe that for any sufficiently small λ > 0 we have that ϕk defined by ϕk (x) = ρφk (x · e)φ (|x|2 /λ 2 ) ∈ V , ∀k ∈ N so that letting k → ∞ we obtain that ϕ(x) = ρφλ (x · e)φ (|x|2 /λ 2 ), is such that (6.11) is satisfied. Moreover, De ϕ(x) = ρφλ′ (x · e)φ (|x|2 /λ 2 ), and Dϕ(x) = ρφλ (x · e)φ ′ (|x|2 /λ 2 )2λ −2 x, where D denotes the gradient relating the variable x. Note that, for such a ϕ(x) the integrand of (6.11) vanishes if x ̸∈ Cλ , where Cλ = {x ∈ Rn : |x · e| ≤ λ and ∥x∥ ≤ λ }. Define Cλ+ and Cλ− by Cλ− = {x ∈ Cλ : x · e ≤ 0}, and Cλ+ = {x ∈ Cλ : x · e > 0}. Hence, denoting gk (x) = ( f (x, u(x), Du(x) + Dϕk (x)) − f (x, u(x), Du(x)) −(ϕk (x) · fs (x, u(x), Du(x) + Dϕk (x) · fξ (x, u(x), Du(x)) (6.13) and g(x) = ( f (x, u(x), Du(x) + Dϕ(x)) − f (x, u(x), Du(x)) −(ϕ(x) · fs (x, u(x), Du(x) + Dϕ(x) · fξ (x, u(x), Du(x))

(6.14)

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letting k → ∞, using the Lebesgue dominated converge theorem we obtain Z

Z

gk (x) dx + −

Cλ+

Cλ

Z

→

Z

g(x) dx + −

Cλ

Cλ+

gk (x) dx

g(x) dx ≥ 0,

(6.15)

Now define y = ye e + y, where ye = and y=

x·e , λ2 x . λ

The sets Cλ− and Cλ+ correspond, concerning the new variables, to the sets B− λ and + Bλ , where −1 ≤ ye ≤ 0}, B− λ = {y : ∥y∥ ≤ 1, and − λ e −1 B+ }. λ = {y : ∥y∥ ≤ 1, and 0 < y ≤ λ

Therefore, since dx = λ n+1 dy, multiplying (6.15) by λ −n−1 , we obtain Z

Z

g(x(y)) dy + −

B1

Z

g(x(y)) dy + −

B− \B1 λ

B+ λ

g(x(y)) dy ≥ 0,

(6.16)

where x = (x · e)e + x = λ 2 ye + λ y ≡ x(y). Observe that ρφ (∥y∥2 ) De ϕ(x) = ρφ (∥y∥2 )(−λ ) 0,

if − 1 ≤ ye ≤ 0, if 0 ≤ ye ≤ λ −1 , otherwise.

Observe also that q |g(x(y))| ≤ o( |ϕ(x)|2 + |Dϕ(x)|2 ), so that from the from the expression of ϕ(x) and Dϕ(x) we obtain, for − − y ∈ B+ λ , or y ∈ Bλ \ B1 ,

that |g(x(y))| ≤ o(λ ), as λ → 0. + Since the Lebesgue measures of B− λ and Bλ are bounded by

2n−1 /λ

(6.17)

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the second and third terms in (6.16) are of o(1) where lim o(1)/λ = 0,

λ →0+

so that letting λ → 0+ , considering that x(y) → 0, and on B− 1 (up to the limit set B) g(x(y))

→

f (0, u(0), Du(0) + ρφ (∥y∥2 )e) − f (0, u(0), Du(0)) − ρφ (∥y∥2 )e fξ (0, u(0), Du(0))

(6.18)

we get, Z

[ f (0, u(0), Du(0) + ρφ (∥y∥2 )e) − f (0, u(0), Du(0))

B

−ρφ (∥y∥2 )e fξ (0, u(0), Du(0))] dy2 ...dyn ≥ 0,

(6.19)

where B is an appropriate limit set (we do not provide more details here) such that B = {y ∈ Rn : ye = 0 and ∥y∥ ≤ 1}. Here we have used the fact that, on the set in question, Dϕ(x) → ρφ (∥y∥2 )e, as λ → 0+ . Finally, inequality (6.19) is valid for a sequence {φn } (in place of φ ) such that 0 ≤ φn ≤ 1 and φn (t) = 1, if |t| < 1 − 1/n, ∀n ∈ N. Letting n → ∞, from (6.19) we obtain f (0, u(0), Du(0) + ρ ⊗ e) − f (0, u(0), Du(0)) −ρ · e fξ (0, u(0), Du(0)) ≥ 0.

6.7

(6.20)

The Weierstrass-Erdmann conditions

We start with a definition. ˆ Definition 6.7.1 Define I = [a, b]. A function u ∈ C([a, b]; RN ) is said to be a weak Lipschitz extremal of Z b

F(u) = a

f (x, u(x), u′ (x)) dx,

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if Z b a

( fs (x, u(x), u′ (x)) · ϕ + fξ (x, u(x), u′ (x)) · ϕ ′ (x)) dx = 0,

∀ϕ ∈ Cc∞ ([a, b]; RN ). Proposition 6.7.2 For any Lipschitz extremal of Z b

F(u) =

f (x, u(x), u′ (x)) dx

a

there exists a constant c ∈ RN such that fξ (x, u(x), u′ (x)) = c + Proof 6.6

Z x a

fs (t, u(t), u′ (t)) dt, ∀x ∈ [a, b].

(6.21)

Fix ϕ ∈ Cc∞ ([a, b]; RN ). Integration by parts of the extremal condition δ F(u, ϕ) = 0,

implies that Z b a

fξ (x, u(x), u′ (x)) · ϕ ′ (x) dx −

Z bZ x a

a

fs (t, u(t), u′ (t)) dt · ϕ ′ (x) dx = 0.

Since ϕ is arbitrary, from the du Bois-Reymond lemma, there exists c ∈ RN such that fξ (x, u(x), u′ (x)) −

Z x a

fs (t, u(t), u′ (t)) dt = c, ∀x ∈ [a, b].

The proof is complete. Theorem 6.7.3 (Weierstrass-Erdmann Corner Conditions) Let I = [a, b]. Suppose u ∈ Cˆ 1 ([a, b]; RN ) is such that F(u) ≤ F(v), ∀v ∈ Cr , for some r > 0. where Cr = {v ∈ Cˆ 1 ([a, b]; RN ) | v(a) = u(a), v(b) = u(b), and ∥u − v∥∞ < r}.

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Let x0 ∈ (a, b) be a corner point of u. Denoting u0 = u(x0 ), ξ0+ = u′ (x0 + 0) and = u′ (x0 − 0), then the following relations are valid:

ξ0−

1. fξ (x0 , u0 , ξ0− ) = fξ (x0 , u0 , ξ0+ ), 2. f (x0 , u0 , ξ0− ) − ξ0− fξ (x0 , u0 , ξ0− ) =

f (x0 , u0 , ξ0+ ) − ξ0+ fξ (x0 , u0 , ξ0+ ).

Remark 6.7.4 The conditions above are known as the Weierstrass-Erdmann corner conditions. Proof 6.7

Condition (1) is just a consequence of equation (6.21). For (2), define τε (x) = x + ελ (x),

where λ ∈ Cc∞ (I). Observe that τε (a) = a and τε (b) = b, ∀ε > 0. Also τ0 (x) = x. Choose ε0 > 0 sufficiently small such that for each ε satisfying |ε| < ε0 , we have τε′ (x) > 0 and u˜ε (x) = (u ◦ τε−1 )(x) ∈ Cr . Define φ (ε) = F(x, u˜ε , u˜′ε (x)). Thus φ has a local minimum at 0, so that φ ′ (0) = 0, that is d(F(x, u˜ε , u˜′ε (x))) |ε=0 = 0. dε Observe that

dτ −1 (x) d u˜ε = u′ (τε−1 (x)) ε , dx dx

and

dτε−1 (x) 1 = . dx 1 + ελ ′ (τε−1 (x))

Thus, Z b

F(u˜ε ) =

a

f x, u(τε−1 (x)), u′ (τε−1 (x))

1 ′ 1 + ελ (τε−1 (x))

Defining x¯ = τε−1 (x), we obtain d x¯ =

1 dx, 1 + ελ ′ (x) ¯

dx.

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that is dx = (1 + ελ ′ (x)) ¯ d x. ¯ Dropping the bar for the new variable, we may write Z b u′ (x) F(u˜ε ) = f x + ελ (x), u(x), 1 + ελ ′ (x) dx. ′ 1 + ελ (x) a From

dF(u˜ε ) |ε=0 , dε

we obtain Z b a

(λ fx (x, u(x), u′ (x)) + λ ′ (x)( f (x, u(x), u′ (x)) − u′ (x) fξ (x, u(x), u′ (x)))) dx = 0. (6.22)

Since λ is arbitrary, from Proposition 6.7.2, we obtain f (x, u(x), u′ (x)) − u′ (x) fξ (x, u(x), u′ (x)) −

Z x a

fx (t, u(t), u′ (t)) dt = c1

for some cR1 ∈ R. Being ax fx (t, u(t), u′ (t)) dt + c1 a continuous function (in fact absolutely continuous), the proof is complete.

6.8

Natural boundary conditions

Consider the functional f : U → R, where Z

f (x, u(x), ∇u(x)) dx,

F(u) Ω

¯ RN , RN×n ), f (x, s, ξ ) ∈ C1 (Ω, and Ω ⊂ Rn is an open bounded connected set. Proposition 6.8.1 Assume U = {u ∈ W 1,2 (Ω; RN ); u = u0 on Γ0 }, where Γ0 ⊂ ∂ Ω is closed and ∂ Ω = Γ = Γ0 ∪ Γ1 being Γ1 open in Γ and Γ0 ∩ Γ1 = 0. / ¯ RN , RN×n ) and u ∈ C2 (Ω; ¯ RN ), and also Thus if ∂ Ω ∈ C1 , f ∈ C2 (Ω, ¯ RN ), such that ϕ = 0 on Γ0 , δ F(u, ϕ) = 0, ∀ϕ ∈ C1 (Ω;

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then u is a extremal of F which satisfies the following natural boundary conditions, nα fxi iα (x, u(x)∇u(x)) = 0, a.e. on Γ1 , ∀i ∈ {1, ..., N}. Proof 6.8 Observe that δ F(u, ϕ) = 0, ∀ϕ ∈ Cc∞ (Ω; RN ), thus u is a extremal of F and through integration by parts and the fundamental lemma of calculus of variations, we obtain L f (u) = 0, in Ω, where L f (u) = fs (x, u(x), ∇u(x)) − div( fξ (x, u(x), ∇u(x)). Defining V = {ϕ ∈ C1 (Ω; RN ) | ϕ = 0 on Γ0 }, for an arbitrary ϕ ∈ V , we obtain Z

L f (u) · ϕ dx

δ F(u, ϕ) = Ω

Z

+ Γ1

Z

= Γ1

nα fxi iα (x, u(x), ∇u(x))ϕ i (x) dΓ

nα fxi iα (x, u(x), ∇u(x))ϕ i (x) dΓ

= 0, ∀ϕ ∈ V .

(6.23)

Suppose, to obtain contradiction, that nα fxi iα (x0 , u(x0 ), ∇u(x0 )) = β > 0, for some x0 ∈ Γ1 and some i ∈ {1, ..., N}. Defining G(x) = nα fxi iα (x, u(x), ∇u(x)), by the continuity of G, there exists r > 0 such that G(x) > β /2, in Br (x0 ), and in particular G(x) > β /2, in Br (x0 ) ∩ Γ1 . Choose 0 < r1 < r such that Br1 (x0 ) ∩ Γ0 = 0. / This is possible since Γ0 is closed and x0 ∈ Γ1 . Choose ϕ i ∈ Cc∞ (Br1 (x0 )) such that ϕ i ≥ 0 in Br1 (x0 ) and ϕ i > 0 in Br1 /2 (x0 ). Therefore Z Z β ϕ i dx > 0, G(x)ϕ i (x) dx > 2 Γ1 Γ1

More Topics on the Calculus of Variations

and this contradicts (6.23). Thus G(x) ≤ 0, ∀x ∈ Γ1 , and by analogy G(x) ≥ 0, ∀x ∈ Γ1 , so that G(x) = 0, ∀x ∈ Γ1 . The proof is complete.

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Chapter 7

Convex Analysis and Duality Theory

7.1

Convex sets and functions

For this section the most relevant reference is Ekeland and Temam [34]. Definition 7.1.1 (Convex Functional) Let U be a vector space and let S ⊂ U be a ¯ = R ∪ {+∞, −∞} is said to be convex, if convex set. A functional F : S → R F(λ u + (1 − λ )v) ≤ λ F(u) + (1 − λ )F(v), ∀u, v ∈ S, λ ∈ [0, 1].

7.2

(7.1)

Weak lower semi-continuity

We start with the definition of Epigraph. ¯ be a funcDefinition 7.2.1 (Epigraph) Let U be a Banach space and let F : U → R tional. We define the Epigraph of F, denoted by E pi(F), by E pi(F) = {(u, a) ∈ U × R | a ≥ F(u)}. Definition 7.2.2 Let U be a Banach space. Consider the weak topology σ (U,U ∗ ) for U and let F : U → R ∪ {+∞} be a functional. Let u ∈ U. We say that F is wekly lower semi-continuous at u ∈ U if for each λ < F(u), there exists a weak neighborhood Vλ (u) ∈ σ (U,U ∗ ) such that F(v) > λ , ∀v ∈ Vλ (u). If F is weakly lower semi-continuous (w.l.s.c.) on U, we write simply F is w.l.s.c.

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Theorem 7.2.3 Let U be a Banach space and let F : U → R ∪ {+∞} be a functional. Under such hypotheses, the following properties are equivalent. 1. F is w.l.s.c. 2. E pi(F) is closed for U × R with the product topology between σ (U,U ∗ ) and the usual topology for R. 3. HγF = {u ∈ U | F(u) ≤ γ} is closed for σ (U,U ∗ ), ∀γ ∈ R. 4. The set GFγ = {u ∈ U | F(u) > γ} is open for σ (U,U ∗ ), ∀γ ∈ R. 5. lim inf F(v) ≥ F(u), ∀u ∈ U, v⇀u

where lim inf F(v) = v⇀u

sup

inf F(v).

V (u)∈σ (U,U ∗ ) v∈V (u)

Proof 7.1 Assume F is w.l.s.c.. We are going to show that E pi(F)c is open for σ (U,U ∗ ) × R. Choose (u, r) ∈ E pi(F)c . Thus (u, r) ̸∈ E pi(F), so that r < F(u). Select λ such that r < λ < F(u). Since F is w.l.s.c. at u, there exists a a weak neighborhood Vλ (u) such that F(v) > λ , ∀v ∈ Vλ (u). Thus, Vλ (u) × (−∞, λ ) ⊂ E pi(F)c so that (u, r) is an interior point of E pi(F)c and hence, since such a point is arbitrary in E pi(F)c , we may infer that E pi(F)c is open so that E pi(F) is closed for the topology in question. Assume now (2). Observe that HγF × {γ} = E pi(F) ∩ (U × {γ}). From the hypotheses E pi(F) is closed, that is, HγF × {γ} is closed and thus HγF is closed. Assume (3). To obtain (4), it suffices to consider the complement of HγF . Suppose (4) is valid. Let u ∈ U and let γ ∈ R be such that γ < F(u). Since GFγ is open for σ (U,U ∗ ) there exists a weak neighborhood V (u) such that V (u) ⊂ GFγ , so that F(v) > γ, ∀v ∈ V (u),

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and hence inf F(v) ≥ γ.

v∈V (u)

In particular, we have lim inf F(v) ≥ γ. v⇀u

Letting γ → F(u), we obtain lim inf F(v) ≥ F(u). v⇀u

Finally assume lim inf F(v) ≥ F(u). v⇀u

Let λ < F(u) and let 0 < ε < F(u) − λ . Observe that lim inf F(v) = sup v⇀u

inf F(v).

V (u)∈σ (U,U ∗ ) v∈V (u)

Thus, there exists a weak neighborhood V (u) such that F(v) ≥ F(u) − ε > λ , ∀v ∈ V (u). The proof is complete. Remark 7.2.4 A similar result is valid for the strong topology (in norm) of a Banach space U so that a F : U → R ∪ {+∞} is strongly lower semi-continuous (s.c.i.) at u ∈ U, if lim inf F(v) ≥ F(u). v→u

(7.2)

Corollary 7.2.5 All convex s.c.i. functional F : U → R is also w.l.s.c. Proof 7.2 The result follows from the fact for F s.c.i, its epigraph being convex and strongly closed, it is also weakly closed. Definition 7.2.6 (Affine-continuous functionals) Let U be a Banach space. A functional F : U → R is said to be affine-continuous, if there exist u∗ ∈ U ∗ and α ∈ R such that F(u) = ⟨u, u∗ ⟩U + α, ∀u ∈ U.

(7.3)

¯ is a funcDefinition 7.2.7 (Γ(U)) Let U be a Banach space. We say that F : U → R tional in Γ(U) and write F ∈ Γ(U) if F may be represented point-wise as the supremum of a family of affine-continuous functionals. If F ∈ Γ(U), and F(u) ∈ R for some u ∈ U we write F ∈ Γ0 (U). ¯ be a Definition 7.2.8 (Convex envelop) Let U be a Banach space. Let F : U → R, ¯ as functional. we define its convex envelop, denoted by CF : U → R, CF(u) =

sup {⟨u, u∗ ⟩ + α}, (u∗ ,α)∈A∗

(7.4)

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where A∗ = {(u∗ , α) ∈ U ∗ × R | ⟨v, u∗ ⟩U + α ≤ F(v), ∀v ∈ U}

7.3

(7.5)

Polar functionals and related topics on convex analysis

¯ Definition 7.3.1 (Polar functional) Let U be a Banach space and let F : U → R, be a functional. We define the polar functional functional related to F, denoted by ¯ by F ∗ : U ∗ → R, F ∗ (u∗ ) = sup{⟨u, u∗ ⟩U − F(u)}, ∀u∗ ∈ U ∗ .

(7.6)

u∈U

Definition 7.3.2 (Bipolar functional) Let U be a Banach space and let F : U → ¯ be a functional. We define the bi-polar functional related to F, denoted by R ¯ as F ∗∗ : U → R, F ∗∗ (u) = sup {⟨u, u∗ ⟩U − F ∗ (u∗ )}, ∀u ∈ U. u∗ ∈U ∗

(7.7)

¯ be a functional. Proposition 7.3.3 Let U be a Banach space and let F : U → R ∗∗ Under such hypotheses F (u) = CF(u), ∀u ∈ U and in particular, if F ∈ Γ(U), then F ∗∗ (u) = F(u), ∀u ∈ U. Proof 7.3 By the definition, the convex envelop of F is the supremum of affinecontinuous functionals bounded by F at the point in question. In fact we need only to consider those which are maximal, that is, only those in the form u 7→ ⟨u, u∗ ⟩U − F ∗ (u∗ ).

(7.8)

CF(u) = sup {⟨u, u∗ ⟩U − F ∗ (u∗ )} = F ∗∗ (u).

(7.9)

Thus, u∗ ∈U ∗

¯ be a functional. Under Corollary 7.3.4 Let U be a Banach space and let F : U → R ∗ ∗∗∗ such hypotheses, F = F . Proof 7.4

Since F ∗∗ ≤ F, we obtain F ∗ ≤ F ∗∗∗ .

(7.10)

F ∗∗ (u) ≥ ⟨u, u∗ ⟩U − F ∗ (u∗ ),

(7.11)

On the other hand,

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so that F ∗∗∗ (u∗ ) = sup{⟨u, u∗ ⟩U − F ∗∗ (u)} ≤ F ∗ (u∗ ).

(7.12)

u∈U

From (7.10) and (7.12), we have F ∗ (u∗ ) = F ∗∗∗ (u∗ ), ∀u∗ ∈ U ∗ . At this point, we recall the definition of Gˆateaux differentiability. Definition 7.3.5 (Gˆateaux differentiability) Let U be a Banach space. A functional ¯ is said to be Gˆateaux differentiable at u ∈ U, if there exists u∗ ∈ U ∗ such F :U →R that lim

λ →0

F(u + λ h) − F(u) = ⟨h, u∗ ⟩U , ∀h ∈ U. λ

(7.13)

The vector u∗ is said to be the Gˆateaux derivative of F : U → R at u and may denoted by u∗ =

∂ F(u) or u∗ = δ F(u) ∂u

(7.14)

¯ be a Definition 7.3.6 (Sub-gradients) Let U be a Banach space and let F : U → R functional. We define the set of sub-gradients of F at u, denoted by ∂ F(u), by ∂ F(u) =

{u∗ ∈ U ∗ , such that ⟨v − u, u∗ ⟩U + F(u) ≤ F(v), ∀v ∈ U}.

(7.15)

Lemma 7.3.7 (Continuity of convex functions) Let U be a Banach space and let F : U → R be a convex functional. Let u ∈ U and suppose there exists a > 0 and a neighborhood V of u such that F(v) < a < +∞, ∀v ∈ V. From the hypotheses, F is continuous at u. Proof 7.5 Redefining the problem with G(v) = F(v + u) − F(u) we need only consider the case in which u = 0 and F(u) = 0. Let V be a neighborhood of 0 such that F(v) ≤ a < +∞, ∀v ∈ V . Define W = V ∩(−V ). Choose ε ∈ (0, 1). Let v ∈ εW , thus v ∈V ε and since F is convex, we have that v F(v) = F (1 − ε)0 + ε ≤ (1 − ε)F(0) + εF(v/ε) ≤ εa. ε

(7.16)

(7.17)

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131

Also −v ∈V. ε

(7.18)

Hence, F(θ ) = F

v (−v/ε) +ε 1+ε 1+ε

≤

F(v) ε + F(−v/ε), 1+ε 1+ε

so that F(v) ≥ (1 + ε)F(θ ) − εF(−v/ε) ≥ −εa.

(7.19)

|F(v)| ≤ εa, ∀v ∈ εW ,

(7.20)

Therefore

that is, F is continuous at u = 0. ¯ be a convex funcProposition 7.3.8 Let U be a Banach space and let F : U → R tional, which is finite and continuous at u ∈ U. Under such hypotheses, ∂ F(u) ̸= 0. / Proof 7.6 Since F is convex, E pi(F) is convex. Since F is continuous at u, we have that E pi(F)0 is non-empty. Observe that (u, F(u)) is on the boundary of E pi(F). Therefore, denoting A = E pi(F), from the Hahn-Banach theorem there exists a closed hyperplane H which separates (u, F(u)) and A0 , where H H = {(v, a) ∈ U × R | ⟨v, u∗ ⟩U + αa = β },

(7.21)

for some fixed α, β ∈ R and u∗ ∈ U ∗ , so that ⟨v, u∗ ⟩U + αa ≥ β , ∀(v, a) ∈ E pi(F),

(7.22)

⟨u, u∗ ⟩U + αF(u) = β ,

(7.23)

and

where (α, β , u∗ ) ̸= (0, 0, 0). Suppose, to obtain contradiction, that α = 0. Thus, ⟨v − u, u∗ ⟩U ≥ 0, ∀v ∈ U,

(7.24)

and therefore we obtain, u∗ = 0 and β = 0, a contradiction. Hence, we may assume α > 0 (considering (7.22)) and thus ∀v ∈ U we have β − ⟨v, u∗ /α⟩U ≤ F(v), α

(7.25)

β − ⟨u, u∗ /α⟩U = F(u), α

(7.26)

and

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that is, ⟨v − u, −u∗ /α⟩U + F(u) ≤ F(v), ∀v ∈ U,

(7.27)

−u∗ /α ∈ ∂ F(u).

(7.28)

so that

The proof is complete. Definition 7.3.9 (Carath´eodory function) Let S ⊂ Rn be an open set. We say that g : S × Rl → R is a Carath´eodory function if ∀ξ ∈ Rl , x 7→ g(x, ξ ) is a measurable function, e a.e. in S, ξ 7→ g(x, ξ ) is a continuous function. The proof of next results may be found in Ekeland and Temam [34]. Proposition 7.3.10 Let E and F be two Banach spaces, let S be a Boreal subset of Rn , and g : S × E → F be a Carath´eodory function. For each measurable function u : S → E, let G1 (u) be the measurable function x 7→ g(x, u(x)) ∈ F. Under such hypotheses, if G1 maps L p (S, E) on Lr (S, F) for 1 ≤ p, r < ∞, then G1 is strongly continuous. For the functional G : U → R, defined by G(u) = U = U ∗ = [L2 (S)]l , we have also the following result.

R

S g(x, u(x))dS,

where

Proposition 7.3.11 Considering the statement in the last proposition we may ex¯ by press G∗ : U ∗ → R ∗

∗

Z

G (u ) =

g∗ (x, u∗ (x))dx,

(7.29)

S

onde g∗ (x, y) = sup (y · η − g(x, η)), for almost all x ∈ S. η∈Rl

7.4

The Legendre transform and the Legendre functional

For non-convex functionals, in some cases, the global extremal through which the polar functional obtained corresponds to a local extremal point at which the analytical expression is possible. This fact motivates the definition of the Legendre Transform, which is obtained through a local extremal point.

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Definition 7.4.1 (Legendre transform and associated functional) Consider the function of C2 class, g : Rn → R. Its Legendre transform, denoted by g∗L : RnL → R, is expressed by n

g∗L (y∗ ) = ∑ x0i · y∗i − g(x0 ),

(7.30)

i=1

where x0 is a solution of the system y∗i =

∂ g(x0 ) , ∂ xi

(7.31)

and RnL = {y∗ ∈ Rn such an equation (7.31) has a unique solution}. R Moreover, considering the functional G : Y → R defined by G(v) = S g(v)dS, we also define the associated Legendre functional, denoted by G∗L : YL∗ → R as G∗L (v∗ ) =

Z S

g∗L (v∗ ) dx,

(7.32)

where YL∗ = {v∗ ∈ Y ∗ | v∗ (x) ∈ RnL , a.e. in S}. About the Legendre transform we still have the following results: Proposition 7.4.2 Considering the last definitions, suppose that for each y∗ ∈ RnL at least in a neighborhood (of y∗ ) it is possible to define a differentiable function by the expression x0 (y∗ ) = [

∂ g −1 ∗ ] (y ). ∂x

(7.33)

Then, ∀ i ∈ {1, ..., n}we may write: y∗i = Proof 7.7

∂ g(x0 ) ∂ g∗ (y∗ ) ⇔ x0i = L ∗ ∂ xi ∂ yi

(7.34)

Suppose firstly that: y∗i =

∂ g(x0 ) , ∀ i ∈ {1, ..., n}, ∂ xi

(7.35)

thus: g∗L (y∗ ) = y∗i x0i − g(x0 )

(7.36)

and taking derivatives for this expression we have: ∂x j ∂ g(x0 ) ∂ x0 j ∂ g∗L (y∗ ) = y∗j 0∗ + x0i − , ∗ ∂ yi ∂ yi ∂ x j ∂ y∗i

(7.37)

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or ∂ g∗L (y∗ ) ∂ g(x0 ) ∂ x0 j = (y∗j − ) ∗ + x0i ∂ y∗i ∂xj ∂ yi

(7.38)

which from (7.35) implies that: ∂ g∗L (y∗ ) = x0i , ∀ i ∈ {1, ..., n}. ∂ y∗i

(7.39)

This completes the first half of the proof. Conversely, suppose now that: x0i =

∂ g∗L (y∗ ) , ∀i ∈ {1, ..., n}. ∂ y∗i

(7.40)

As y∗ ∈ RnL there exists x¯0 ∈ Rn such that: y∗i =

∂ g(x¯0 ) ∀i ∈ {1, ..., n}, ∂ xi

(7.41)

and, g∗L (y∗ ) = y∗i x¯0i − g(x¯0 )

(7.42)

and therefore taking derivatives for this expression we can obtain: ∂ x¯ j ∂ g(x¯0 ) ∂ x¯0 j ∂ g∗L (y∗ ) = y∗j 0∗ + x¯0i − , ∂ y∗i ∂ yi ∂ x j ∂ y∗i

(7.43)

∀ i ∈ {1, ..., n}, so that: ∂ g∗L (y∗ ) ∂ g(x¯0 ) ∂ x¯0 j = (y∗j − ) ∗ + x¯0i ∂ y∗i ∂xj ∂ yi

(7.44)

∀ i ∈ {1, ..., n}, which from (7.40) and (7.41), implies that: x¯0i =

∂ g∗L (y∗ ) = x0i , ∀ i ∈ {1, ..., n}, ∂ y∗i

(7.45)

from this and (7.41) we have: y∗i =

∂ g(x¯0 ) ∂ g(x0 ) = ∀ i ∈ {1, ..., n}. ∂ xi ∂ xi

(7.46)

¯ defined as J(u) = (G ◦ Λ)(u) − Theorem 7.4.3 Consider the functional J : U → R ⟨u, f ⟩U where Λ(= {Λi }) : U → Y (i ∈ {1, ..., n}) is a continuous linear operator R and, G : Y → R is a functional that can be expressed as G(v) = S g(v)dS, ∀v ∈ Y (here g : Rn → R is a differentiable function that admits Legendre Transform denoted by g∗L : RnL → R. That is, the hypothesis mentioned at Proposition 7.4.2 are satisfied).

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Under these assumptions we have: δ J(u0 ) = θ ⇔ δ (−G∗L (v∗0 ) + ⟨u0 , Λ∗ v∗0 − f ⟩U ) = θ , where v∗0 = case:

∂ G(Λ(u0 )) ∂v

is supposed to be such that v∗0 (x) ∈ RnL , a.e. in S and in this J(u0 ) = −G∗L (v∗0 ).

Proof 7.8

(7.47)

(7.48)

Suppose first that δ J(u0 ) = θ , that is: Λ∗

which, as v∗0 =

∂ G(Λu0 ) ∂v

∂ G(Λu0 ) −f =θ ∂v

(7.49)

implies that: Λ∗ v∗0 − f = θ ,

(7.50)

∂ g(Λu0 ) . ∂ xi

(7.51)

and v∗0i =

Thus from the last proposition we can write: Λi (u0 ) =

∂ g∗L (v∗0 ) , for i ∈ {1, .., n} ∂ y∗i

(7.52)

which means: Λu0 =

∂ G∗L (v∗0 ) . ∂ v∗

(7.53)

Therefore from (7.50) and (7.53) we have: δ (−G∗L (v∗0 ) + ⟨u0 , Λ∗ v∗0 − f ⟩U ) = θ .

(7.54)

This completes the first part of the proof. Conversely, suppose now that: δ (−G∗L (v∗0 ) + ⟨u0 , Λ∗ v∗0 − f ⟩U ) = θ ,

(7.55)

Λ∗ v∗0 − f = θ

(7.56)

that is:

and Λu0 =

∂ G∗L (v∗0 ) . ∂ v∗

(7.57)

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Clearly, from (7.57), the last proposition and (7.56) we can write: v∗0 =

∂ G(Λ(u0 )) ∂v

(7.58)

and Λ∗

∂ G(Λu0 ) − f = θ, ∂v

(7.59)

which implies: δ J(u0 ) = θ .

(7.60)

J(u0 ) = G(Λu0 ) − ⟨u0 , f ⟩U

(7.61)

Finally, we have:

From this, (7.56) and (7.58) we have J(u0 ) = G(Λu0 ) − ⟨u0 , Λ∗ v∗0 ⟩U = G(Λu0 ) − ⟨Λu0 , v∗0 ⟩Y = −G∗L (v∗0 ). (7.62)

7.5

Duality in convex optimization

¯ (F ∈ Γ0 (U)) we define the problem Let U be a Banach space. Given F : U → R P as P : minimize F(u) on U.

(7.63)

We say that u0 ∈ U is a solution of problem P if F(u0 ) = infu∈U F(u). Consider a ¯ such that function φ (u, p) (φ : U ×Y → R) φ (u, 0) = F(u),

(7.64)

P ∗ : maximize − φ ∗ (0, p∗ ) on Y ∗ .

(7.65)

we define the problem P ∗ , as

Observe that φ ∗ (0, p∗ ) =

sup

{⟨0, u⟩U + ⟨p, p∗ ⟩Y − φ (u, p)} ≥ −φ (u, 0),

(7.66)

(u,p)∈U×Y

or inf {φ (u, 0)} ≥ sup {−φ ∗ (0, p∗ )}.

u∈U

p∗ ∈Y ∗

(7.67)

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Proposition 7.5.1 Consider φ ∈ Γ0 (U ×Y ). If we define h(p) = inf {φ (u, p)},

(7.68)

u∈U

then h is convex. Proof 7.9

We have to show that given p, q ∈ Y and λ ∈ (0, 1), we have h(λ p + (1 − λ )q) ≤ λ h(p) + (1 − λ )h(q).

(7.69)

If h(p) = +∞ or h(q) = +∞ we are done. Thus let us assume h(p) < +∞ and h(q) < +∞. For each a > h(p) there exists u ∈ U such that h(p) ≤ φ (u, p) ≤ a,

(7.70)

and, if b > h(q), there exists v ∈ U such that h(q) ≤ φ (v, q) ≤ b.

(7.71)

Thus h(λ p + (1 − λ )q) ≤ inf {φ (w, λ p + (1 − λ )q)} w∈U

≤ φ (λ u + (1 − λ )v, λ p + (1 − λ )q) ≤ λ φ (u, p) + (1 − λ )φ (v, q) ≤ λ a + (1 − λ )b. (7.72) Letting a → h(p) and b → h(q) we obtain h(λ p + (1 − λ )q) ≤ λ h(p) + (1 − λ )h(q). □

(7.73)

Proposition 7.5.2 For h as above, we have h∗ (p∗ ) = φ ∗ (0, p∗ ), ∀p∗ ∈ Y ∗ , so that h∗∗ (0) = sup {−φ ∗ (0, p∗ )}. p∗ ∈Y ∗

(7.74)

Proof. Observe that h∗ (p∗ ) = sup{⟨p, p∗ ⟩Y − h(p)} = sup{⟨p, p∗ ⟩Y − inf {φ (u, p)}}, p∈Y

p∈Y

u∈U

(7.75)

so that h∗ (p∗ ) =

sup

{⟨p, p∗ ⟩Y − φ (u, p)} = φ ∗ (0, p∗ ).

(7.76)

(u,p)∈U×Y

Proposition 7.5.3 The set of solutions of the problem P ∗ (the dual problem) is identical to ∂ h∗∗ (0).

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Proof 7.10

Consider p∗0 ∈ Y ∗ a solution of Problem P ∗ , that is, −φ ∗ (0, p∗0 ) ≥ −φ ∗ (0, p∗ ), ∀p∗ ∈ Y ∗ ,

(7.77)

−h∗ (p∗0 ) ≥ −h∗ (p∗ ), ∀p∗ ∈ Y ∗ ,

(7.78)

which is equivalent to

which is equivalent to − h(p∗0 ) = sup {⟨0, p∗ ⟩Y − h∗ (p∗ )} ⇔ −h∗ (p∗0 ) = h∗∗ (0) p∗ ∈Y ∗

⇔ p∗0 ∈ ∂ h∗∗ (0). (7.79) ¯ convex. Assume infu∈U {φ (u, 0)} ∈ R and Theorem 7.5.4 Consider φ : U × Y → R there exists u0 ∈ U such that p 7→ φ (u0 , p) is finite and continuous at 0 ∈ Y , then inf {φ (u, 0)} = sup {−φ ∗ (0, p∗ )},

u∈U

p∗ ∈Y ∗

(7.80)

and the dual problem has at least one solution. Proof 7.11 By hypothesis h(0) ∈ R and as was shown above, h is convex. As the function p 7→ φ (u0 , p) is convex and continuous at 0 ∈ Y , there exists a neighborhood V of zero in Y such that φ (u0 , p) ≤ M < +∞, ∀p ∈ V ,

(7.81)

for some M ∈ R. Thus, we may write h(p) = inf {φ (u, p)} ≤ φ (u0 , p) ≤ M, ∀p ∈ V . u∈U

(7.82)

Hence, from Lemma 7.3.7, h is continuous at 0. Thus by Proposition 7.3.8, h is subdifferentiable at 0, which means h(0) = h∗∗ (0). Therefore by Proposition 7.5.3, the dual problem has solutions and h(0) = inf {φ (u, 0)} = sup {−φ ∗ (0, p∗ )} = h∗∗ (0). u∈U

p∗ ∈Y ∗

(7.83)

Now we apply the last results to φ (u, p) = G(Λu + p) + F(u), where Λ : U → Y is a continuous linear operator whose adjoint operator is denoted by Λ∗ : Y ∗ → U ∗ . We may enunciate the following theorem. Theorem 7.5.5 Suppose U is a reflexive Banach space and define J : U → R by J(u) = G(Λu) + F(u) = φ (u, 0),

(7.84)

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where lim J(u) = +∞ as ∥u∥U → ∞ and F ∈ Γ0 (U), G ∈ Γ0 (Y ). Also suppose there exists uˆ ∈ U such that J(u) ˆ < +∞ with the function p 7→ G(p) continuous at Λu. ˆ Under such hypothesis, there exist u0 ∈ U and p∗0 ∈ Y ∗ such that J(u0 ) = min{J(u)} = max {−G∗ (p∗ ) − F ∗ (−Λ∗ p∗ )} ∗ ∗ p ∈Y

u∈U

= −G∗ (p∗0 ) − F ∗ (−Λ∗ p∗0 ). (7.85) Proof 7.12 The existence of solutions for the primal problem follows from the direct method of calculus of variations. That is, considering a minimizing sequence, from above (coercivity hypothesis), such a sequence is bounded and has a weakly convergent subsequence to some u0 ∈ U. Finally, from the lower semi-continuity of primal formulation, we may conclude that u0 is a minimizer. The other conclusions follow from Theorem 7.5.4, observing that φ ∗ (0, p∗ ) =

sup {⟨p, p∗ ⟩Y − G(Λu + p) − F(u)} u∈U,p∈Y

=

sup {⟨q, p∗ ⟩ − G(q) − ⟨Λu, p∗ ⟩ − F(u)}, (7.86) u∈U,q∈Y

so that φ ∗ (0, p∗ ) = G∗ (p∗ ) + sup{−⟨u, Λ∗ p∗ ⟩U − F(u)} u∈U

= G∗ (p∗ ) + F ∗ (−Λ∗ p∗ ). (7.87) Thus, inf {φ (u, 0)} = sup {−φ ∗ (0, p∗ )}

u∈U

p∗ ∈Y ∗

(7.88)

and solutions u0 and p∗0 for the primal and dual problems, respectively, imply that J(u0 ) = min{J(u)} = max {−G∗ (p∗ ) − F ∗ (−Λ∗ p∗ )} ∗ ∗ u∈U

p ∈Y

= −G∗ (p∗0 ) − F ∗ (−Λ∗ p∗0 ). (7.89)

7.6

The min-max theorem

Our main objective in this section is to state and prove the min-max theorem. Definition 7.1 Let U,Y be Banach spaces, A ⊂ U and B ⊂ Y and let L : A × B → R be a functional. We say that (u0 , v0 ) ∈ A × B is a saddle point for L if L(u0 , v) ≤ L(u0 , v0 ) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B.

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Proposition 7.1 Let U,Y be Banach spaces, A ⊂ U and B ⊂ Y . A functional L : U × Y → R has a saddle point if and only if max inf L(u, v) = min sup L(u, v). v∈B u∈A

Proof 7.13 Thus,

u∈A v∈B

Suppose (u0 , v0 ) ∈ A × B is a saddle point of L.

L(u0 , v) ≤ L(u0 , v0 ) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B.

(7.90)

Define F(u) = sup L(u, v). v∈B

Observe that inf F(u) ≤ F(u0 ),

u∈A

so that inf sup L(u, v) ≤ sup L(u0 , v).

u∈A v∈B

(7.91)

v∈B

Define G(v) = inf L(u, v). u∈A

Thus sup G(v) ≥ G(v0 ), v∈B

so that sup inf L(u, v) ≥ inf L(u, v0 ). u∈A

v∈B u∈A

(7.92)

From (7.90), (7.91) and (7.92) we obtain inf sup L(u, v) ≤ sup L(u0 , v)

u∈A v∈B

v∈B

≤ L(u0 , v0 ) ≤ inf L(u, v0 ) u∈A

≤ sup inf L(u, v). v∈B u∈A

(7.93)

Hence inf sup L(u, v) ≤ L(u0 , v0 )

u∈A v∈B

≤ sup inf L(u, v). v∈B u∈A

(7.94)

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On the other hand inf L(u, v) ≤ L(u, v), ∀u ∈ A, v ∈ B,

u∈A

so that sup inf L(u, v) ≤ sup L(u, v), ∀u ∈ A, v∈B u∈A

vInB

and hence sup inf L(u, v) ≤ inf sup L(u, v). u∈A v∈B

v∈B u∈A

(7.95)

From (7.90), (7.94), (7.95) we obtain inf sup L(u, v) =

sup L(u0 , v)

u∈A v∈B

v∈B

= L(u0 , v0 ) = inf L(u, v0 ) u∈A

= sup inf L(u, v). v∈B u∈A

(7.96)

Conversely suppose max inf L(u, v) = min sup L(u, v). v∈B u∈A

u∈A v∈B

As above defined, F(u) = sup L(u, v), v∈B

and G(v) = inf L(u, v). u∈A

From the hypotheses, there exists (u0 , v0 ) ∈ A × B such that sup G(v) = G(v0 ) = F(u0 ) = inf F(u). u∈A

v∈B

so that F(u0 ) = sup L(u0 , v) = inf L(u, v0 ) = G(v0 ). v∈B

u∈U

In particular L(u0 , v0 ) ≤ sup L(u0 , v) = inf L(u, v0 ) ≤ L(u0 , v0 ). v∈B

u∈U

Therefore sup L(u0 , v) = L(u0 , v0 ) = inf L(u, v0 ). v∈B

u∈U

The proof is complete. Proposition 7.2 Let U,Y be Banach spaces, A ⊂ U, B ⊂ Y and let L : A × B → R be a functional. Assume there exist u0 ∈ A, v0 ∈ B and α ∈ R such that

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L(u0 , v) ≤ α, ∀v ∈ B, and L(u, v0 ) ≥ α, ∀u ∈ A. Under such hypotheses (u0 , v0 ) is a saddle point of L, that is, L(u0 , v) ≤ L(u0 , v0 ) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B. Proof 7.14

Observe, from the hypotheses we have L(u0 , v0 ) ≤ α,

and L(u0 , v0 ) ≥ α, so that L(u0 , v) ≤ α = L(u0 , v0 ) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B. This completes the proof. In the next lines we state and prove the min − max theorem. Theorem 7.1 Let U,Y be reflexive Banach spaces, A ⊂ U, B ⊂ Y and let L : A × B → R be a functional. Suppose that 1. A ⊂ U is convex, closed and non-empty. 2. B ⊂ Y is convex, closed and non-empty. 3. For each u ∈ A, Fu (v) = L(u, v) is concave and upper semi-continuous. 4. For each v ∈ B, Gv (u) = L(u, v) is convex and lower semi-continuous. 5. The set A and B are bounded. Under such hypotheses L has at least one saddle point (u0 , v0 ) ∈ A × B such that L(u0 , v0 ) =

min max L(u, v) u∈A v∈B

= max min L(u, v). v∈B u∈A

(7.97)

Proof 7.15 Fix v ∈ B. Observe that Gv (u) = L(u, v) is convex and lower semicontinuous, therefore it is weakly lower semi-continuous on the weak compact set A. At first we assume the additional hypothesis that Gv (u) is strictly convex, ∀v ∈ B. Hence Fv (u) attains a unique minimum on A. We denote the optimal u ∈ A by u(v)

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Define G(v) = min Gv (u) = min L(u, v). u∈U

u∈A

Thus, G(v) = L(u(v), v). The function G(v) is expressed as the minimum of a family of concave weakly upper semi-continuous functions, and hence it is also concave and upper semicontinuous. Moreover, G(v) is bounded above on the weakly compact set B, so that there exists v0 ∈ B such that G(v0 ) = max G(v) = max min L(u, v). v∈B

v∈B u∈A

Observe that G(v0 ) = min L(u, v0 ) ≤ L(u, v0 ), ∀u ∈ U. u∈A

Observe that, from the concerned concavity, for u ∈ A, v ∈ B and λ ∈ (0, 1) we have L(u, (1 − λ )v0 + λ v) ≥ (1 − λ )L(u, v0 ) + λ L(u, v). In particular denote u((1 − λ )v0 + λ v) = uλ , where uλ is such that G((1 − λ )v0 + λ v) =

min L(u, (1 − λ )v0 + λ v) u∈A

= L(uλ , (1 − λ )v0 + λ v).

(7.98)

Therefore, G(v0 ) = ≥ = ≥ ≥

max G(v) v∈B

G((1 − λ )v0 + λ v) L(uλ , (1 − λ )v0 + λ v) (1 − λ )L(uλ , v0 ) + λ L(uλ , v) (1 − λ ) min L(u, v0 ) + λ L(uλ , v) u∈A

= (1 − λ )G(v0 ) + λ L(uλ , v).

(7.99)

From this, we obtain G(v0 ) ≥ L(uλ , v).

(7.100)

Let {λn } ⊂ (0, 1) be such that λn → 0. Let {un } ⊂ A be such that G((1 − λn )v0 + λn v) =

min L(u, (1 − λn )v0 + λn v) u∈A

= L(un , (1 − λn )v0 + λn v).

(7.101)

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Since A is weakly compact, there exists a subsequence {unk } ⊂ {un } ⊂ A and u0 ∈ A such that unk ⇀ u0 , weakly in U, as k → ∞. Observe that (1 − λnk )L(unk , v0 ) + λnk L(unk , v) ≤ L(unk , (1 − λnk )v0 + λnk v) = min L(u, (1 − λnk )v0 + λnk v) u∈A

≤ L(u, (1 − λnk )v0 + λnk v),

(7.102)

∀u ∈ A, k ∈ N. Recalling that λnk → 0, from this and (7.102) we obtain L(u0 , v0 ) ≤ lim inf L(unk , v0 ) k→∞

= lim inf((1 − λnk )L(unk , v0 ) + λnk L(u, v)) k→∞

≤ lim sup L(u, (1 − λnk )v0 + λnk v) k→∞

≤ L(u, v0 ), ∀u ∈ U.

(7.103)

Hence, L(u0 , v0 ) = minu∈A L(u, v0 ). Observe that from (7.100) we have G(v0 ) ≥ L(unk , v), so that G(v0 ) ≥ lim inf L(unk , v) ≥ L(u0 , v), ∀v ∈ B. k→∞

Denoting α = G(v0 ) we have α = G(v0 ) ≥ L(u0 , v), ∀v ∈ B, and α = G(v0 ) = min L(u, v0 ) ≤ L(u, v0 ), ∀u ∈ A. u∈U

From these last two results and Proposition 7.2 we have that (u0 , v0 ) is a saddle point for L. Now assume that Gv (u) = L(u, v) is convex but not strictly convex ∀v ∈ B. For each n ∈ N Define Ln by Ln (u, v) = L(u, v) + ∥u∥U /n.

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In such a case (Fv )n (u) = Ln (u, v) is strictly convex for all n ∈ N. From above we main obtain (un , vn ) ∈ A × B such that L(un , v) + ∥un ∥U /n ≤ L(un , vn ) + ∥un ∥U /n ≤ L(u, vn ) + ∥u∥/n.

(7.104)

Since A × B is weakly compact and {(un , vn )} ⊂ A × B, up to subsequence not relabeled, there exists (u0 , v0 ) ∈ A × B such that un ⇀ u0 , weakly in U, vn ⇀ v0 , weakly in Y, so that, L(u0 , v) ≤ lim inf(L(un , v) + ∥un ∥U /n) n→∞

≤ lim sup L(u, vn ) + ∥u∥U /n n→∞

≤ L(u, v0 ).

(7.105)

Hence, L(u0 , v) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B, so that L(u0 , v) ≤ L(u0 , v0 ) ≤ L(u, v0 ), ∀u ∈ A, v ∈ B. This completes the proof. In the next result we deal with more general situations. Theorem 7.2 Let U,Y be reflexive Banach spaces, A ⊂ U, B ⊂ Y and let L : A × B → R be a functional. Suppose that 1. A ⊂ U is convex, closed and non-empty. 2. B ⊂ Y is convex, closed and non-empty. 3. For each u ∈ A, Fu (v) = L(u, v) is concave and upper semi-continuous. 4. For each v ∈ B, Gv (u) = L(u, v) is convex and lower semi-continuous. 5. Either the set A is bounded or there exists v˜ ∈ B such that L(u, v) ˜ → +∞, as ∥u∥ → +∞, u ∈ A.

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6. Either the set B is bounded or there exists u˜ ∈ A such that L(u, ˜ v) → −∞, as ∥v∥ → +∞, v ∈ B. Under such hypotheses L has at least one saddle point (u0 , v0 ) ∈ A × B. Proof 7.16 such that

We prove the result just for the special case such that there exists v˜ ∈ B L(u, v) ˜ → +∞, as ∥u∥ → +∞, u ∈ A,

and B is bounded. The proofs of remaining cases are similar. For each n ∈ N denote An = {u ∈ A : ∥u∥U ≤ n}. Fix n ∈ N. The sets An and B are closed, convex and bounded, so that from the last Theorem 7.1 there exists a saddle point (un , vn ) ∈ An × B for L : An × B → R. Hence, L(un , v) ≤ L(un , vn ) ≤ L(u, vn ), ∀u ∈ An , v ∈ B. For a fixed u˜ ∈ A1 we have L(un , v) ˜ ≤ L(un , vn ) ≤ L(u, ˜ vn ) ≤ sup L(u, ˜ v) ≡ b ∈ R.

(7.106)

v∈B

On the other hand, from the hypotheses, Gv˜ (u) = L(u, v) ˜ is convex, lower semi-continuous and coercive, so that it is bounded below. Thus there exists a ∈ R such that −∞ < a ≤ Fv˜ (u) = L(u, v), ˜ ∀u ∈ A. Hence a ≤ L(un , v) ˜ ≤ L(un , vn ) ≤ b, ∀n ∈ N. Therefore {L(un , vn )} is bounded. Moreover, from the coercivity hypotheses and a ≤ L(un , v) ˜ ≤ b, ∀n ∈ N, we may infer that {un } is bounded.

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Summarizing, {un }, {vn }, and {L(un , vn )} are bounded sequences, and thus there exists a subsequence {nk }, u0 ∈ A, v0 ∈ B and α ∈ R such that unk ⇀ u0 , weakly in U, vnk ⇀ v0 , weakly in Y, L(unk , vnk ) → α ∈ R, as k → ∞. Fix (u, v) ∈ A × B. Observe that if nk > n0 = ∥u∥U , then L(unk , v) ≤ L(unk , vnk ) ≤ L(u, vnk ), so that letting k → ∞, we obtain L(u0 , v) ≤ lim inf L(unk , v) k→∞

≤

lim L(unk , vnk ) = α

k→∞

≤ lim sup L(u, vnk ) k→∞

≤ L(u, v0 ),

(7.107)

that is, L(u0 , v) ≤ α ≤ L(u, v0 ), ∀u ∈ A, v ∈ B. From this and Proposition 7.2 we may conclude that (u0 , v0 ) is a saddle point for L : A × B → R. The proof is complete.

7.7

Relaxation for the scalar case

In this section, Ω ⊂ RN denotes a bounded open set with a locally Lipschitz boundary. That is, for each point x ∈ ∂ Ω there exists a neighborhood Ux whose the intersection with ∂ Ω is the graph of a Lipschitz continuous function. We start with the following definition. Definition 7.7.1 A function u : Ω → R is said to be affine if ∇u is constant on Ω. Furthermore, we say that u : Ω → R is piecewise affine if it is continuous and there exists a partition of Ω into a set of zero measure and finite number of open sets on which u is affine. The proof of next result is found in [34]. Theorem 7.7.2 Let r ∈ N and let uk , 1 ≤ k ≤ r be piecewise affine functions from Ω into R and {αk } such that αk > 0, ∀k ∈ {1, ..., r} and ∑rk=1 αk = 1. Given ε > 0, there exists a locally Lipschitz function u : Ω → R and r disjoint open sets Ωk , 1 ≤ k ≤ r, such that |m(Ωk ) − αk m(Ω)| < αk ε, ∀k ∈ {1, ..., r},

(7.108)

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∇u(x) = ∇uk (x), a.e. on Ωk ,

(7.109)

|∇u(x)| ≤ max {|∇uk (x)|}, a.e. on Ω,

(7.110)

r u(x) − ∑ αk uk < ε, ∀x ∈ Ω, k=1

(7.111)

1≤k≤r

r

u(x) =

∑ αk uk (x), ∀x ∈ ∂ Ω.

(7.112)

k=1

The next result is also found in [34]. Proposition 7.7.3 Let r ∈ N and let uk , 1 ≤ k ≤ r be piecewise affine functions from Ω into R. Consider a Carath´eodory function f : Ω × RN → R and a positive function c ∈ L1 (Ω) which satisfy c(x) ≥ sup{| f (x, ξ )| | |ξ | ≤ max {∥∇uk ∥∞ }}. 1≤k≤r

Given ε > 0, there exists a locally Lipschitz function u : Ω → R such that Z Z r f (x, ∇uk )dx < ε, f (x, ∇u)dx − ∑ αk Ω Ω k=1 |∇u(x)| ≤ max {|∇uk (x)|}, a.e. in Ω, 1≤k≤r

(7.113)

(7.114)

(7.115)

r

|u(x) − ∑ αk uk (x)| < ε, ∀x ∈ Ω

(7.116)

k=1

r

u(x) =

∑ αk uk (x), ∀x ∈ ∂ Ω.

(7.117)

k=1

Proof 7.17 It is sufficient to establish the result for functions uk affine over Ω, since Ω can be divided into pieces on which uk are affine, and such pieces can be put together through (7.117). Let ε > 0 be given. We know that simple functions are dense in L1 (Ω), concerning the L1 norm. Thus there exists a partition of Ω into a finite number of open sets Oi , 1 ≤ i ≤ N1 and a negligible set, and there exists f¯k constant functions over each Oi such that Z Ω

| f (x, ∇uk (x)) − f¯k (x)|dx < ε, 1 ≤ k ≤ r.

(7.118)

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Now choose δ > 0 such that δ≤

ε N1 (1 + max1≤k≤r {∥ f¯k ∥∞ })

(7.119)

and if B is a measurable set m(B) < δ ⇒

Z B

c(x)dx ≤ ε/N1 .

(7.120)

Now we apply Theorem 7.7.2, to each of the open sets Oi , therefore there exists a locally Lipschitz function u : Oi → R and there exist r open disjoints spaces Ωik , 1 ≤ k ≤ r, such that |m(Ωik ) − αk m(Oi )| ≤ αk δ , for 1 ≤ k ≤ r,

(7.121)

∇u = ∇uk , a.e. in Ωik ,

(7.122)

|∇u(x)| ≤ max {|∇uk (x)|}, a.e. Oi ,

(7.123)

r u(x) − ∑ αk uk (x) ≤ δ , ∀x ∈ Oi k=1

(7.124)

1≤k≤r

r

u(x) =

∑ αk uk (x), ∀x ∈ ∂ Oi .

(7.125)

k=1 1 We can define u = ∑rk=1 αk uk on Ω − ∪Ni=1 Oi . Therefore u is continuous and locally Lipschitz. Now observe that

Z Oi

r

f (x, ∇u(x))dx − ∑

Z

i k=1 Ωk

f (x, ∇uk (x))dx Z

=

Oi −∪rk=1 Ωik

f (x, ∇u(x))dx. (7.126)

From | f (x, ∇u(x))| ≤ c(x), m(Oi − ∪rk=1 Ωik ) ≤ δ and (7.120) we obtain Z r Z f (x, ∇u(x))dx − ∑ f (x, ∇uk (x)dx i Oi Ω k=1 k Z = f (x, ∇u(x))dx ≤ ε/N1 . (7.127) i r O −∪ Ω i

k=1

k

Considering that f¯k is constant in Oi , from (7.119), (7.120) and (7.121) we obtain r

∑| k=1

Z Ωik

f¯k (x)dx − αk

Z Oi

f¯k (x)dx| < ε/N1 .

(7.128)

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1 We recall that Ωk = ∪Ni=1 Ωik so that Z Z r f (x, ∇uk (x))dx f (x, ∇u(x))dx − ∑ αk Ω Ω k=1 Z r Z f (x, ∇uk (x))dx ≤ f (x, ∇u(x))dx − ∑ Ω Ω k k=1

r

Z

| f (x, ∇uk (x) − f¯k (x)|dx

+∑

k=1 Ωk r Z

+ ∑ k=1 r

Ωk

+ ∑ αk k=1

f¯k (x)dx − αk

Z Ω

Z

f¯k (x)dx

| f¯k (x) − f (x, ∇uk (x))|dx.

(7.129)

Ω

From (7.127), (7.118), (7.128) and (7.118) again, we obtain Z Z r f (x, ∇uk )dx < 4ε. f (x, ∇u(x))dx − ∑ αk Ω Ω k=1

(7.130)

The next result we do not prove it. It is a well known result from the finite element theory. Proposition 7.7.4 If u ∈ W01,p (Ω) there exists a sequence {un } of piecewise affine functions over Ω, null on ∂ Ω, such that un → u, in L p (Ω)

(7.131)

∇un → ∇u, in L p (Ω; RN ).

(7.132)

and

Proposition 7.7.5 For p such that 1 < p < ∞, suppose that f : Ω × RN → R is a Carath´eodory function , for which there exist a1 , a2 ∈ L1 (Ω) and constants c1 ≥ c2 > 0 such that a2 (x) + c2 |ξ | p ≤ f (x, ξ ) ≤ a1 (x) + c1 |ξ | p , ∀x ∈ Ω, ξ ∈ RN .

(7.133)

Then, given u ∈ W 1,p (Ω) piecewise affine, ε > 0 and a neighborhood V of zero in the topology σ (L p (Ω, RN ), Lq (Ω, RN )) there exists a function v ∈ W 1,p (Ω) such that ∇v − ∇u ∈ V ,

(7.134)

u = v on ∂ Ω, ∥v − u∥∞ < ε,

(7.135)

Z Z ∗∗ f (x, ∇v(x))dx − f (x, ∇u(x))dx < ε. Ω Ω

(7.136)

and

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Proof 7.18 Suppose given ε > 0, u ∈ W 1,p (Ω) piecewise affine continuous, and a neighborhood V of zero, which may be expressed as Z V = {w ∈ L (Ω, R ) | hm · wdx < η, Ω p

N

∀m ∈ {1, ..., M}}, (7.137) where M ∈ N, hm ∈ Lq (Ω, RN ), η ∈ R+ . By hypothesis, there exists a partition of Ω into a negligible set Ω0 and open subspaces ∆i , 1 ≤ i ≤ r, over which ∇u(x) is constant. From standard results of convex analysis in RN , for each i ∈ {1, ..., r} we can obtain {αk ≥ 0}1≤k≤N+1 , and ξk such that ∑N+1 k=1 αk = 1 and N+1

∑ αk ξk = ∇u, ∀x ∈ ∆i ,

(7.138)

k=1

and N+1

∑ αk f (x, ξk ) = f ∗∗ (x, ∇u(x)).

(7.139)

k=1

Define βi = maxk∈{1,...,N+1} {|ξk | on ∆i }, and ρ1 = maxi∈{1,...,r} {βi }, and ρ = max{ρ1 , ∥∇u∥∞ }. Now, observe that we can obtain functions hˆ m ∈ C0∞ (Ω; RN ) such that η . (7.140) max ∥hˆ m − hm ∥Lq (Ω,RN ) < 4ρm(Ω) m∈{1,...,M} Define C = maxm∈{1,...,M} ∥div(hˆ m )∥Lq (Ω) and we can also define ε1 = min{ε/4, 1/(m(Ω)1/p ), η/(2Cm(Ω)1/p ), 1/m(Ω)}

(7.141)

We recall that ρ does not depend on ε. Furthermore, for each i ∈ {1, ..., r} there exists a compact subset Ki ⊂ ∆i such that Z ∆i −Ki

[a1 (x) + c1 (x) max {|ξ | p }]dx < |ξ |≤ρ

ε1 . r

(7.142)

Also, observe that the sets Ki may be obtained such that the restrictions of f and f ∗∗ to Ki × ρB are continuous, so that from this and from the compactness of ρB, for all x ∈ Ki , we can find an open ball ωx with center in x and contained in Ω, such that | f ∗∗ (y, ∇u(x)) − f ∗∗ (x, ∇u(x))|

0. Assume for some u ∈ U we have F(u) ≤ inf {F(u)} + ε. u∈U

Under such hypotheses, there exists v ∈ U such that 1. d(u, v) ≤ 1, 2. F(v) ≤ F(u), 3. F(v) ≤ F(w) + εd(v, w), ∀w ∈ U.

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The Method of Lines and Duality Principles for Non-Convex Models

Proof 7.22

Define the sequence {un } ⊂ U by: u1 = u,

and having u1 , ..., un , select un+1 as specified in the next lines. First, define Sn = {w ∈ U | F(w) ≤ F(un ) − εd(un , w)}. Observe that un ∈ Sn so that Sn in non-empty. On the other hand, from the definition of infimum, we may select un+1 ∈ Sn such that 1 F(un ) + inf {F(w)} . (7.175) F(un+1 ) ≤ w∈Sn 2 Since un+1 ∈ Sn we have εd(un+1 , un ) ≤ F(un ) − F(un+1 ).

(7.176)

and hence m

εd(un+m , un ) ≤ ∑ d(un+i , un+i−1 ) ≤ F(un ) − F(un+m ).

(7.177)

i=1

From (7.176) {F(un )} is decreasing sequence bounded below by infu∈U F(u) so that there exists α ∈ R such that F(un ) → α as n → ∞. From this and (7.177), {un } is a Cauchy sequence , converging to some v ∈ U. Since F is lower semi-continuous we get, α = lim inf F(un+m ) ≥ F(v), m→∞

so that letting m → ∞ in (7.177) we obtain εd(un , v) ≤ F(un ) − F(v), and, in particular for n = 1 we get 0 ≤ εd(u, v) ≤ F(u) − F(v) ≤ F(u) − inf F(u) ≤ ε. u∈U

Thus, we have proven 1 and 2. Suppose, to obtain contradiction, that 3 does not hold. Hence, there exists w ∈ U such that F(w) < F(v) − εd(w, v).

(7.178)

Convex Analysis and Duality Theory

■

157

In particular, we have w ̸= v.

(7.179)

Thus, from this and (7.178) we have F(w) < F(un ) − ε(un , v) − εd(w, v) ≤ F(un ) − εd(un , w), ∀n ∈ N. Now observe that w ∈ Sn , ∀n ∈ N so that inf {F(w)} ≤ F(w), ∀n ∈ N.

w∈Sn

From this and (7.175) we obtain, 2F(un+1 ) − F(un ) ≤ F(w) < F(v) − εd(v, w), so that 2 lim inf{F(un+1 )} ≤ F(v) − εd(v, w) + lim inf{F(un )}. n→∞

n→∞

Hence, F(v) ≤ lim inf{F(un+1 )} ≤ F(v) − εd(v, w), n→∞

so that 0 ≤ −εd(v, w), which contradicts (7.179). Thus 3 holds. Remark 7.8.2 We may introduce in U a new metric given by d1 = ε 1/2 d. We highlight that the topology remains the same and also F remains lower semi-continuous. Under the hypotheses of the last theorem, if there exists u ∈ U such that F(u) < infu∈U F(u) + ε, then there exists v ∈ U such that 1. d(u, v) ≤ ε 1/2 , 2. F(v) ≤ F(u), 3. F(v) ≤ F(w) + ε 1/2 d(u, w), ∀w ∈ U. Remark 7.8.3 Observe that, if U is a Banach space, F(v) − F(v + tw) ≤ ε 1/2t∥w∥U , ∀t ∈ [0, 1], w ∈ U,

(7.180)

so that, if F is Gˆateaux differentiable, we obtain −⟨δ F(v), w⟩U ≤ ε 1/2 ∥w∥U .

(7.181)

Similarly F(v) − F(v + t(−w)) ≤ ε 1/2t∥w∥U ≤, ∀t ∈ [0, 1], w ∈ U,

(7.182)

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The Method of Lines and Duality Principles for Non-Convex Models

so that, if F is Gˆateaux differentiable, we obtain ⟨δ F(v), w⟩U ≤ ε 1/2 ∥w∥U .

(7.183)

∥δ F(v)∥U ∗ ≤ ε 1/2 .

(7.184)

Thus

We have thus obtained, from the last theorem and remarks, the following result. Theorem 7.8.4 Let U be a Banach space. Let F : U → R be a lower semi-continuous Gˆateaux differentiable functional. Given ε > 0 suppose that u ∈ U is such that F(u) ≤ inf {F(u)} + ε. u∈U

(7.185)

Then there exists v ∈ U such that F(v) ≤ F(u), ∥u − v∥U ≤

√ ε,

(7.186) (7.187)

and ∥δ F(v)∥U ∗ ≤

√ ε.

(7.188)

The next theorem easily follows from above results. Theorem 7.8.5 Let J : U → R, be defined by J(u) = G(∇u) − ⟨ f , u⟩L2 (S;RN ) ,

(7.189)

U = W01,2 (S; RN ),

(7.190)

where

We suppose G is a l.s.c and Gˆateaux-differentiable so that J is bounded below. Then, given ε > 0, there exists uε ∈ U such that J(uε ) < inf {J(u)} + ε,

(7.191)

√ ε. □

(7.192)

u∈U

and ∥δ J(uε )∥U ∗

0, there exists n0 ∈ N such that if n ≥ n0 then G∗ (v∗0 + z∗0 ) − ⟨Λun , v∗0 ⟩Y − ⟨Λun , z∗0 ⟩Y + G(Λun ) < ε/2. On the other hand, since F(Λ1 u) is convex and l.s.c. we have lim sup{−F(Λ1 un )} ≤ −F(Λ1 u0 ). n→∞

Hence, there exists n1 ∈ N such that if n ≥ n1 then ⟨Λun , z∗0 ⟩Y − F(Λ1 un ) ≤ ⟨Λu0 , z∗0 ⟩Y − F(Λ1 u0 ) +

ε ε = F ∗ (L∗ z∗0 ) + , 2 2

so that for all n ≥ max{n0 , n1 } we obtain G∗ (v∗0 + z∗0 ) − F ∗ (L∗ z∗0 ) − ⟨un , f ⟩U − F(Λ1 un ) + G(Λun ) < ε. Since ε is arbitrary, the proof is complete.

7.9

Some examples of duality theory in convex and non-convex analysis

We start with a well known result of Toland, published in 1979. Theorem 7.9.1 (Toland, 1979) Let U be a Banach space and let F, G : U → R be functionals such that inf {G(u) − F(u)} = α ∈ R. u∈U

Under such hypotheses F ∗ (u∗ ) − G∗ (u∗ ) ≥ α, ∀u∗ ∈ U ∗ . Moreover, suppose that u0 ∈ U is such that G(u0 ) − F(u0 ) = min{G(u) − F(u)} = α. u∈U

u∗0

Assume also ∈ ∂ F(u0 ). Under such hypotheses, F ∗ (u∗0 ) − G∗ (u∗0 ) = α, so that G(u0 ) − F(u0 ) = = =

min{G(u) − F(u)} u∈U

min {F ∗ (u∗ ) − G∗ (u∗ )} u∗ ∈U ∗ F ∗ (u∗0 ) − G∗ (u∗0 ).

(7.193)

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Proof 7.24

Under such hypotheses, inf {G(u) − F(u)} = α ∈ R.

u∈U

Thus G(u) − F(u) ≥ α, ∀u ∈ U. Therefore, for u∗ ∈ U ∗ , we have −⟨u, u∗ ⟩U + G(u) + ⟨u, u∗ ⟩U − F(u) ≥ α, ∀u ∈ U. Thus, −⟨u, u∗ ⟩U + G(u) + sup{⟨u, u∗ ⟩U − F(u)} ≥ α, ∀u ∈ U, u∈U

that is, −⟨u, u∗ ⟩U + G(u) + F ∗ (u∗ ) ≥ α, ∀u ∈ U, so that inf {−⟨u, u∗ ⟩U + G(u)} + F ∗ (u∗ ) ≥ α,

u∈U

that is, −G∗ (u∗ ) + F ∗ (u∗ ) ≥ α, ∀u∗ ∈ U ∗ .

(7.194)

Also from the hypotheses, G(u0 ) − F(u0 ) ≤ G(u) − F(u), ∀u ∈ U. On the other hand, from

u∗0

(7.195)

∈ ∂ F(u0 ), we obtain

⟨u0 , u∗0 ⟩U − F(u0 ) ≥ ⟨u, u∗0 ⟩U − F(u), ∀u ∈ U so that −F(u) ≤ ⟨u0 − u, u∗0 ⟩U − F(u0 ), ∀u ∈ U. From this and (7.195), we get, G(u0 ) − F(u0 ) ≤ G(u) + ⟨u0 − u, u∗0 ⟩U − F(u0 ), ∀u ∈ U.

(7.196)

so that, ⟨u0 , u∗0 ⟩U − G(u0 ) ≥ ⟨u, u∗0 ⟩U − G(u), ∀u ∈ U, that is G∗ (u∗0 ) =

sup{⟨u, u∗0 ⟩U − G(u)}

u∈U

= ⟨u0 , u∗0 ⟩U − G(u0 ).

(7.197)

Convex Analysis and Duality Theory

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163

Summarizing, we have got G∗ (u∗0 ) = ⟨u0 , u∗0 ⟩U − G(u0 ), and F ∗ (u∗0 ) = ⟨u0 , u∗0 ⟩U − F(u0 ). Hence, F ∗ (u0 ) − G∗ (u∗0 ) = G(u0 ) − F(u0 ) = α. From this and (7.194), we have G(u0 ) − F(u0 ) =

min{G(u) − F(u)} u∈U

min {F ∗ (u∗ ) − G∗ (u∗ )} u∗ ∈U ∗ F ∗ (u∗0 ) − G∗ (u∗0 ).

= =

(7.198)

The proof is complete. Exercise 7.9.2 Let Ω ⊂ R2 be a set of Cˆ 1 class. Let V = C1 (Ω) and let J : D ⊂ V → R where Z Z γ ∇u · ∇u dx − f u dx, ∀u ∈ U J(u) = 2 Ω Ω and where D = {u ∈ V : u = 0 on ∂ Ω}. 1. Prove that J is convex. 2. Prove that u0 ∈ D such that γ∇2 u0 + f = 0, in Ω, minimizes J on D. 3. Prove that inf J(u) ≥ sup {−G∗ (v∗ ) − F ∗ (−Λ∗ v∗ )},

u∈U

v∗ ∈Y ∗

where G(∇u) =

1 2

Z

∇u · ∇u dx,

Ω

and G∗ (v∗ ) = sup{⟨v, v∗ ⟩Y − G(v)}, v∈Y

where Y = Y ∗ = L2 (Ω).

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The Method of Lines and Duality Principles for Non-Convex Models

■

4. Defining Λ : U → Y by Λu = ∇u, and F :D→R as

Z

F(u) =

f u dx, Ω

so that F ∗ (−Λ∗ v∗ ) =

sup{−⟨∇u, v∗ ⟩Y − F(u)} Z ∗ f u dx = sup ⟨u, div v ⟩Y + Ω u∈D Z ∗ = sup (div v + f ) u dx Ω u∈D 0, if div(v∗ ) + f = 0, in Ω = +∞, otherwise, u∈D

(7.199)

prove que v∗0 = γ∇u0 is such that J(u0 ) =

min J(u) u∈D

= min{G(Λu) + F(u)} u∈D

=

max {−G∗ (v∗ ) − F ∗ (−Λ∗ v∗ )}

v∗ ∈Y ∗ ∗

= −G (v∗0 ) − F ∗ (−Λ∗ v∗0 ).

(7.200)

Solution: Let u ∈ D and v ∈ Va = {v ∈ V : v = 0 on ∂ Ω}. Thus,

= = =

δ J(u; v) J(u + εv) − J(u) lim ε ε→0 R R R (γ/2) Ω (∇u + ε∇v) · (∇u + ε∇v) dx − (γ/2) Ω ∇u · ∇u dx − Ω (u + εv − u) f dx lim ε ε→0 Z Z Z lim γ ∇u · ∇v dx − f v dx + ε(γ/2) ∇v · ∇v dx ε→0

Z

=

γ Ω

Ω

∇u · ∇v dx −

Ω

Ω

Z

f v dx. Ω

(7.201)

Convex Analysis and Duality Theory

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165

Hence, J(u + v) − J(u) = (γ/2)

Z

(∇u + ∇v) · (∇u + ∇v) dx − (γ/2)

Z

Ω

−

Z

(u + v − u) f dx

ZΩ

∇u · ∇v dx −

= γ ZΩ

≥ γ

∇u · ∇u dx

Ω

∇u · ∇v dx −

Z

Z

∇v · ∇v dx

f v dx + (γ/2) ZΩ

Ω

f v dx

Ω

Ω

= δ J(u; v)

(7.202)

∀u ∈ D, v ∈ Va . From this we may infer that J is convex. From the hypotheses u0 ∈ D is such that γ∇2 u0 + f = 0, in Ω. Let v ∈ Va . Therefore, we have Z

δ J(u0 ; v) =

∇u0 · ∇v dx −

γ ZΩ

= γ

Z

∇u0 · ∇v dx + γ

Ω

Z

Z

∇2 u0 v dx

Ω

∇u0 · ∇v dx − γ

= γ

f v dx Ω

Ω

Z

∇u0 · ∇v dx +

Ω

Z

∇u0 · n v ds

∂Ω

= 0

(7.203)

where n denotes the unit outward field to ∂ Ω. Summarizing, we got δ J(u0 ; v) = 0, ∀v ∈ Va . Since J is convex,from this we may conclude that u0 minimizes J on D. Observe now that, J(u) = G(∇u) + F(u) = −⟨∇u, v∗ ⟩Y + G(∇u) + ⟨∇u · v∗ ⟩Y + F(u) ≥ inf {−⟨v, v∗ ⟩Y + G(v)} v∈Y

+ inf {⟨∇u, v∗ ⟩Y + F(u)} u∈U

= −G∗ (v∗ ) − F ∗ (−Λ∗ v∗ ), ∀u ∈ U, v∗ ∈ Y ∗ .

(7.204)

Summarizing, inf J(u) ≥ sup {−G∗ (v∗ ) − F ∗ (−Λ∗ v∗ )}.

u∈D

Also from the hypotheses we Thus,

v∗ ∈Y ∗ have v∗0 =

v∗0 =

γ∇u0 . ∂ G(∇u0 ) , ∂v

(7.205)

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The Method of Lines and Duality Principles for Non-Convex Models

so that G∗ (v∗0 ) =

sup{⟨v, v∗0 ⟩Y − G(v)} v∈Y

= ⟨∇u0 , v∗0 ⟩Y − G(∇u0 ) = −⟨u0 , div v∗0 ⟩L2 − G(∇u0 ).

(7.206)

On the other hand, from de v∗0 = γ∇u0 , we have div v∗0 = γdiv(∇u0 ) = γ∇2 u0 = − f From this and (7.206), we obtain, G∗ (v∗0 ) = −⟨u0 , f ⟩L2 − G(∇u0 ), and from div v∗0 + f = 0 we get F ∗ (−Λ∗ v∗0 ) = 0. hence G(∇u0 ) − ⟨u0 , f ⟩L2 = −G∗ (v∗0 ) − F ∗ (−Λ∗ v∗0 ), so that from this and (7.205) we have J(u0 ) =

min J(u) u∈D

= min{G(Λu) + F(u)} u∈D

=

max {−G∗ (v∗ ) − F ∗ (−Λ∗ v∗ )}

v∗ ∈Y ∗ ∗

= −G (v∗0 ) − F ∗ (−Λ∗ v∗0 ). A solution is complete.

(7.207)

Chapter 8

Constrained Variational Optimization

8.1

Basic concepts

For this chapter, the most relevant reference is the excellent book of Luenberger, [55], where more details may be found. We start with the definition of cone: Definition 8.1.1 (Cone) Given U a Banach space, we say that C ⊂ U is a cone with vertex at origin, if given u ∈ C, we have that λ u ∈ C, ∀λ ≥ 0. By analogy we define a cone with vertex at p ∈ U as P = p +C, where C is any cone with vertex at origin. Definition 8.1.2 Let P be a convex cone in U. For u, v ∈ U we write u ≥ v (with respect to P) if u − v ∈ P. In particular u ≥ θ if and only if u ∈ P. Also P+ = {u∗ ∈ U ∗ | ⟨u, u∗ ⟩U ≥ 0, ∀u ∈ P}.

(8.1)

If u∗ ∈ P+ we write u∗ ≥ θ ∗ . Proposition 8.1.3 Let U be a Banach space and P be a convex closed cone in U. If u ∈ U satisfies ⟨u, u∗ ⟩U ≥ 0, ∀u∗ ≥ θ ∗ , then u ≥ θ . Proof 8.1 We prove the contrapositive. Assume u ̸∈ P. Then by the separating hyperplane theorem there is an u∗ ∈ U ∗ such that ⟨u, u∗ ⟩U < ⟨p, u∗ ⟩U , ∀p ∈ P. Since P is cone we must have ⟨p, u∗ ⟩U ≥ 0, otherwise we would have ⟨u, u∗ ⟩ > ⟨α p, u∗ ⟩U for some α > 0. Thus u∗ ∈ P+ . Finally, since inf p∈P {⟨p, u∗ ⟩U } = 0, we obtain ⟨u, u∗ ⟩U < 0 which completes the proof.

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Definition 8.1.4 (Convex Mapping) Let U, Z be vector spaces. Let P ⊂ Z be a convex cone. A mapping G : U → Z is said to be convex if the domain of G is convex and G(αu1 + (1 − α)u2 ) ≤ αG(u1 ) + (1 − α)G(u2 ), ∀u1 , u2 ∈ U, α ∈ [0, 1]. (8.2) Consider the problem P, defined as Problem P : Minimize F : U → R subject to u ∈ Ω, and G(u) ≤ θ Define ω(z) = inf{F(u) | u ∈ Ω and G(u) ≤ z}.

(8.3)

For such a functional we have the following result. Proposition 8.1.5 If F is a real convex functional and G is convex, then ω is convex. Proof 8.2

Observe that

ω(αz1 + (1 − α)z2 ) = inf{F(u) | u ∈ Ω and G(u) ≤ αz1 + (1 − α)z2 } (8.4) ≤ inf{F(u) | u = αu1 + (1 − α)u2 u1 , u2 ∈ Ω and G(u1 ) ≤ z1 , G(u2 ) ≤ z2 } (8.5) ≤α inf{F(u1 ) | u1 ∈ Ω, G(u1 ) ≤ z1 } + (1 − α) inf{F(u2 ) | u2 ∈ Ω, G(u2 ) ≤ z2 } ≤αω(z1 ) + (1 − α)ω(z2 ).

(8.6) (8.7)

Now we establish the Lagrange multiplier theorem for convex global optimization. Theorem 8.1.6 Let U be a vector space, Z a Banach space, Ω a convex subset of U, P a positive convex closed cone of Z. Assume that P contains an interior point. Let F be a real convex functional on Ω and G a convex mapping from Ω into Z. Assume the existence of u1 ∈ Ω such that G(u1 ) < θ . Defining µ0 = inf {F(u) | G(u) ≤ θ }, u∈Ω

(8.8)

then there exists z∗0 ≥ θ , z∗0 ∈ Z ∗ such that µ0 = inf {F(u) + ⟨G(u), z∗0 ⟩Z }. u∈Ω

(8.9)

Furthermore, if the infimum in (8.8) is attained by u0 ∈ U such that G(u0 ) ≤ θ , it is also attained in (8.9) by the same u0 and also ⟨G(u0 ), z∗0 ⟩Z = 0. We refer to z∗0 as the Lagrangian Multiplier.

Constrained Variational Optimization

Proof 8.3

■

169

Consider the space W = R × Z and the sets A, B where

A = {(r, z) ∈ R × Z | r ≥ F(u), z ≥ G(u) f or some u ∈ Ω},

(8.10)

and B = {(r, z) ∈ R × Z | r ≤ µ0 , z ≤ θ },

(8.11)

where µ0 = infu∈Ω {F(u) | G(u) ≤ θ }. Since F and G are convex, A and B are convex sets. It is clear that A contains no interior point of B, and since N = −P contains an interior point , the set B contains an interior point. Thus, from the separating hyperplane theorem, there is a non-zero element w∗0 = (r0 , z∗0 ) ∈ W ∗ such that r0 r1 + ⟨z1 , z∗0 ⟩Z ≥ r0 r2 + ⟨z2 , z∗0 ⟩Z , ∀(r1 , z1 ) ∈ A, (r2 , z2 ) ∈ B.

(8.12)

From the nature of B it is clear that w∗0 ≥ θ . That is, r0 ≥ 0 and z∗0 ≥ θ . We will show that r0 > 0. The point (µ0 , θ ) ∈ B, hence r0 r + ⟨z, z∗0 ⟩Z ≥ r0 µ0 , ∀(r, z) ∈ A.

(8.13)

If r0 = 0 then ⟨G(u1 ), z∗0 ⟩Z ≥ 0 and z∗0 ̸= θ . Since G(u1 ) < θ and z∗0 ≥ θ we have a contradiction. Therefore r0 > 0 and, without loss of generality we may assume r0 = 1. Since the point (µ0 , θ ) is arbitrarily close to A and B, we have µ0 = inf {r + ⟨z, z∗0 ⟩Z } ≤ inf {F(u) + ⟨G(u), z∗0 ⟩Z } (r,z)∈A

u∈Ω

≤ inf{F(u) | u ∈ Ω, G(u) ≤ θ } = µ0 . (8.14) Also, if there exists u0 such that G(u0 ) ≤ θ , µ0 = F(u0 ), then µ0 ≤ F(u0 ) + ⟨G(u0 ), z∗0 ⟩Z ≤ F(u0 ) = µ0 .

(8.15)

⟨G(u0 ), z∗0 ⟩Z = 0.

(8.16)

Hence

Corollary 8.1.7 Let the hypothesis of the last theorem hold. Suppose F(u0 ) = inf {F(u) | G(u) ≤ θ }. u∈Ω

(8.17)

Then there exists z∗0 ≥ θ such that the Lagrangian L : U × Z ∗ → R defined by L(u, z∗ ) = F(u) + ⟨G(u), z∗ ⟩Z

(8.18)

has a saddle point at (u0 , z∗0 ). That is L(u0 , z∗ ) ≤ L(u0 , z∗0 ) ≤ L(u, z∗0 ), ∀u ∈ Ω, z∗ ≥ θ .

(8.19)

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Proof 8.4

For z∗0 obtained in the last theorem, we have L(u0 , z∗0 ) ≤ L(u, z∗0 ), ∀u ∈ Ω.

As

⟨G(u0 ), z∗0 ⟩Z

(8.20)

= 0, we have

L(u0 , z∗ ) − L(u0 , z∗0 ) = ⟨G(u0 ), z∗ ⟩Z − ⟨G(u0 ), z∗0 ⟩Z = ⟨G(u0 ), z∗ ⟩Z ≤ 0. (8.21) We now prove two theorems relevant to develop the subsequent section. Theorem 8.1.8 Let F : Ω ⊂ U → R and G : Ω → Z. Let P ⊂ Z be a convex closed cone. Suppose there exists (u0 , z∗0 ) ∈ U × Z ∗ where z∗0 ≥ θ and u0 ∈ Ω are such that F(u0 ) + ⟨G(u0 ), z∗0 ⟩Z ≤ F(u) + ⟨G(u), z∗0 ⟩Z , ∀u ∈ Ω.

(8.22)

Then F(u0 ) + ⟨G(u0 ), z∗0 ⟩Z = inf{F(u) | u ∈ Ω and G(u) ≤ G(u0 )}. (8.23) Proof 8.5 Thus

Suppose there is a u1 ∈ Ω such that F(u1 ) < F(u0 ) and G(u1 ) ≤ G(u0 ). ⟨G(u1 ), z∗0 ⟩Z ≤ ⟨G(u0 ), z∗0 ⟩Z

(8.24)

F(u1 ) + ⟨G(u1 ), z∗0 ⟩Z < F(u0 ) + ⟨G(u0 ), z∗0 ⟩Z ,

(8.25)

so that

which contradicts the hypothesis of the theorem. Theorem 8.1.9 Let F be a convex real functional and G : Ω → Z convex and let u0 and u1 be solutions to the problems P0 and P1 respectively, where P0 : minimize F(u) subject to u ∈ Ω and G(u) ≤ z0 ,

(8.26)

P1 : minimize F(u) subject to u ∈ Ω and G(u) ≤ z1 .

(8.27)

and Suppose z∗0 and z∗1 are the Lagrange multipliers related to these problems. Then ⟨z1 − z0 , z∗1 ⟩Z ≤ F(u0 ) − F(u1 ) ≤ ⟨z1 − z0 , z∗0 ⟩Z . Proof 8.6

(8.28)

For u0 , z∗0 we have

F(u0 ) + ⟨G(u0 ) − z0 , z∗0 ⟩Z ≤ F(u) + ⟨G(u) − z0 , z∗0 ⟩Z , ∀u ∈ Ω, and,

particularly for u = u1 and considering that ⟨G(u0 ) − z0 , z∗0 ⟩Z = F(u0 ) − F(u1 ) ≤ ⟨G(u1 ) − z0 , z∗0 ⟩Z ≤ ⟨z1 − z0 , z∗0 ⟩Z .

A similar argument applied to

u1 , z∗1

provides us the other inequality.

(8.29)

0, we obtain (8.30)

Constrained Variational Optimization

8.2

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171

Duality

Consider the basic convex programming problem: Minimize F(u) subject to G(u) ≤ θ , u ∈ Ω,

(8.31)

where F : U → R is a convex functional, G : U → Z is convex mapping, and Ω is a convex set. We define ϕ : Z ∗ → R by ϕ(z∗ ) = inf {F(u) + ⟨G(u), z∗ ⟩Z }. u∈Ω

(8.32)

Proposition 8.2.1 ϕ is concave and ϕ(z∗ ) = inf {ω(z) + ⟨z, z∗ ⟩Z },

(8.33)

ω(z) = inf {F(u) | G(u) ≤ z},

(8.34)

z∈Γ

where u∈Ω

and Γ = {z ∈ Z | G(u) ≤ z f or some u ∈ Ω}. Proof 8.7

Observe that ϕ(z∗ ) = ≤

inf {F(u) + ⟨G(u), z∗ ⟩Z }

u∈Ω

inf {F(u) + ⟨z, z∗ ⟩Z | G(u) ≤ z}

u∈Ω

= ω(z) + ⟨z, z∗ ⟩Z , ∀z∗ ≥ θ , z ∈ Γ.

(8.35)

On the other hand, for any u1 ∈ Ω, defining z1 = G(u1 ), we obtain F(u1 ) + ⟨G(u1 ), z∗ ⟩Z ≥ inf {F(u) + ⟨z1 , z∗ ⟩Z | G(u) ≤ z1 } u∈Ω

= ω(z1 ) + ⟨z1 , z∗ ⟩Z , (8.36) so that ϕ(z∗ ) ≥ inf {ω(z) + ⟨z, z∗ ⟩Z }. z∈Γ

(8.37)

Theorem 8.2.2 (Lagrange duality) Consider F : Ω ⊂ U → R a convex functional, Ω a convex set, and G : U → Z a convex mapping. Suppose there exists a u1 such that G(u1 ) < θ and that infu∈Ω {F(u) | G(u) ≤ θ } < ∞, with such order related to a convex closed cone in Z. Under such assumptions, we have inf {F(u) | G(u) ≤ θ } = max {ϕ(z∗ )}. ∗ z ≥θ

u∈Ω

(8.38)

If the infimum on the left side in (8.38) is achieved at some u0 ∈ U and the max on the right side at z∗0 ∈ Z ∗ , then ⟨G(u0 ), z∗0 ⟩Z = 0 and u0 minimizes

F(u) + ⟨G(u), z∗0 ⟩Z

on Ω.

(8.39)

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Proof 8.8

For z∗ ≥ θ we have

inf {F(u) + ⟨G(u), z∗ ⟩Z } ≤

u∈Ω

inf u∈Ω,G(u)≤θ

{F(u) + ⟨G(u), z∗ ⟩Z } ≤

inf u∈Ω,G(u)≤θ

F(u) ≤ µ0 . (8.40)

or ϕ(z∗ ) ≤ µ0 .

(8.41)

The result follows from Theorem 8.1.6.

8.3

The Lagrange multiplier theorem

Remark 8.3.1 This section was published in similar form by the journal “Computational and Applied Mathematics, SBMAC-Springer”, reference [24]. In this section, we develop a new and simpler proof of the Lagrange multiplier theorem in a Banach space context. In particular, we address the problem of minimizing a functional F : U → R subject to G(u) = θ , where θ denotes the zero vector and G : U → Z is a Fr´echet differentiable transformation. Here U, Z are Banach spaces. General results on Banach spaces may be found in [13, 35] for example. For the theorem in question, among others we would cite [55, 51, 22]. Specially the proof given in [55] is made through the generalized inverse function theorem. We emphasize such a proof is extensive and requires the continuous Fr´echet differentiability of F and G. Our approach here is different and the results are obtained through other hypotheses. The main result is summarized by the following theorem. Theorem 8.3.2 Let U and Z be Banach spaces. Assume u0 is a local minimum of F(u) subject to G(u) = θ , where F : U → R is a Gˆateaux differentiable functional and G : U → Z is a Fr´echet differentiable transformation such that G′ (u0 ) maps U onto Z. Finally, assume there exist α > 0 and K > 0 such that if ∥ϕ∥U < α then, ∥G′ (u0 + ϕ) − G′ (u0 )∥ ≤ K∥ϕ∥U . Under such assumptions, there exists z∗0 ∈ Z ∗ such that F ′ (u0 ) + [G′ (u0 )]∗ (z∗0 ) = θ , that is, ⟨ϕ, F ′ (u0 )⟩U + ⟨G′ (u0 )ϕ, z∗0 ⟩Z = 0, ∀ϕ ∈ U.

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Proof 8.9 First observe that there is no loss of generality in assuming 0 < α < 1. Also from the generalized mean value inequality and our hypothesis, if ∥ϕ∥U < α, then ∥G(u0 + ϕ) − G(u0 ) − G′ (u0 ) · ϕ∥ ≤ sup ∥G′ (u0 + hϕ) − G′ (u0 )∥ ∥ϕ∥U h∈[0,1]

≤ K sup {∥hϕ∥U }∥ϕ∥U ≤ K∥ϕ∥U2 .

(8.42)

h∈[0,1]

For each ϕ ∈ U, define H(ϕ) by G(u0 + ϕ) = G(u0 ) + G′ (u0 ) · ϕ + H(ϕ), that is, H(ϕ) = G(u0 + ϕ) − G(u0 ) − G′ (u0 ) · ϕ. Let L0 = N(G′ (u0 )) where N(G′ (u0 )) denotes the null space of G′ (u0 ). Observe that U/L0 is a Banach space for which we define A : U/L0 → Z by A(u) ¯ = G′ (u0 ) · u, where u¯ = {u + v | v ∈ L0 }. Since G′ (u0 ) is onto, so is A, so that by the inverse mapping theorem A has a continuous inverse A−1 . α , Let ϕ ∈ U be such that G′ (u0 ) · ϕ = θ . For a given t such that 0 < |t| < 1+∥ϕ∥ U let ψ0 ∈ U be such that H(tϕ) G′ (u0 ) · ψ0 + 2 = θ , t Observe that, from (8.42), ∥H(tϕ)∥ ≤ Kt 2 ∥ϕ∥U2 , and thus from the boundedness of A−1 , ∥ψ0 ∥ as a function of t may be chosen uniformly bounded relating t (that is, despite the fact that ψ0 may vary with t, there α ). exists K1 > 0 such that ∥ψ0 ∥U < K1 , ∀t such that 0 < |t| < 1+∥ϕ∥ U Now choose 0 < r < 1/4 and define g0 = θ . Also define r ε= . −1 4(∥A ∥ + 1)(K + 1)(K1 + 1)(∥ϕ∥U + 1) Since from the hypotheses G′ (u) is continuous at u0 , we may choose 0 < δ < α such that if ∥v∥U < δ then ∥G′ (u0 + v) − G′ (u0 )∥ < ε. Fix t ∈ R such that 0 < |t|

0 such that φ (0) = F(u0 ) ≤ F(u0 + tϕ + t 2 ψ˜ 0 (t)) = φ (t), ∀|t| < t˜2 , also from the hypothesis we get φ ′ (0) = δ F(u0 , ϕ) = 0, that is, ⟨ϕ, F ′ (u0 )⟩U = 0, ∀ϕ such that G′ (u0 ) · ϕ = θ . In the next lines as usual N[G′ (u0 )] and R[G′ (u0 )] denote the null space and the range of G′ (u0 ), respectively. Thus F ′ (u0 ) is orthogonal to the null space of G′ (u0 ), which we denote by F ′ (u0 ) ⊥ N[G′ (u0 )]. Since R[G′ (u0 )] is closed, we get F ′ (u0 ) ∈ R([G′ (u0 )]∗ ), that is, there exists z∗0 ∈ Z ∗ such that F ′ (u0 ) = [G′ (u0 )]∗ (−z∗0 ). The proof is complete.

8.4

Some examples concerning inequality constraints

In this section, we assume the hypotheses of last theorem for F and G below specified. As an application of this same result, consider the problem of locally minimizing F(u) subject to G1 (u) = θ and G2 (u) ≤ θ , where F : U → R, U being a function Banach space, G1 : U → [L p (Ω)]m1 , G2 : U → [L p (Ω)]m2 where 1 < p < ∞, and Ω is an appropriate subset of RN . We refer to the simpler case in which the partial order in [L p (Ω)]m2 is defined by u = {ui } ≥ θ if and only if ui ∈ L p (Ω) and ui (x) ≥ 0 a.e. in Ω, ∀i ∈ {1, ..., m2 }. Observe that defining ˜ v) = F(u), F(u, G(u, v) = {(G1 )i (u)}m1 ×1 , {(G2 )i (u) + v2i }m2 ×1 ˜ v) subject to G(u, v) = (θ , θ ) is equivalent it is clear that (locally) minimizing F(u, to the original problem. We clarify the domain of F˜ is denoted by U ×Y , where Y = {v measurable such that v2i ∈ L p (Ω), ∀i ∈ {1, ..., m2 }}. Therefore, if u0 is a local minimum for the original constrained problem, then for an appropriate and easily defined v0 we have that (u0 , v0 ) is a point of local minimum for the extended constrained one, so that by the last theorem there exists a Lagrange multiplier z∗0 = (z∗1 , z∗2 ) ∈ [Lq (Ω)]m1 × [Lq (Ω)]m2 where 1/p + 1/q = 1 and F˜ ′ (u0 , v0 ) + [G′ (u0 , v0 )]∗ (z∗0 ) = (θ , θ ),

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that is, F ′ (u0 ) + [G′1 (u0 )]∗ (z∗1 ) + [G′2 (u0 )]∗ (z∗2 ) = θ ,

(8.50)

and (z∗2 )i v0i = θ , ∀i ∈ {1, ..., m2 }. In particular for almost all x ∈ Ω, if x is such that v0i (x)2 > 0 then z∗2i (x) = 0, and if v0i (x) = 0 then (G2 )i (u0 (x)) = 0, so that (z∗2 )i (G2 )i (u0 ) = 0, a.e. in Ω, ∀i ∈ {1, ..., m2 }. ˜ 0 , v) = F(u0 ) subFurthermore, consider the problem of minimizing F1 (v) = F(u ject {G2i (u0 ) + v2i } = θ . From above such a local minimum is attained at v0 . Thus, from the stationarity of F1 (v) + ⟨z∗2 , {(G2 )i (u0 ) + v2i }⟩[L p (Ω)]m2 at v0 and the standard necessary conditions for the case of convex (in fact quadratic) constraints we get (z∗2 )i ≥ 0 a.e. in Ω, ∀i ∈ {1, ..., m2 }, that is, z∗2 ≥ θ . Summarizing, for the order in question the first order necessary optimality conditions are given by (8.50), z∗2 ≥ θ and (z∗2 )i (G2 )i (u0 ) = θ , ∀i ∈ {1, ..., m2 } (so that ⟨z∗2 , G2 (u0 )⟩[L p (Ω)]m2 = 0), G1 (u0 ) = θ , and G2 (u0 ) ≤ θ . Remark 8.4.1 For the case U = Rn and Rmk replacing [L p (Ω)]mk , for k ∈ {1, 2} the conditions (z∗2 )i vi = θ means that for the constraints not active (for example vi ̸= 0) the corresponding coordinate (z∗2 )i of the Lagrange multiplier is 0. If vi = 0 then (G2 )i (u0 ) = 0, so that in any case (z∗2 )i (G2 )i (u0 ) = 0. Summarizing, for this last mentioned case we have obtained the standard necessary optimality conditions: (z∗2 )i ≥ 0, and (z∗2 )i (G2 )i (u0 ) = 0, ∀i ∈ {1, ..., m2 }.

8.5

The Lagrange multiplier theorem for equality and inequality constraints

In this section, we develop more rigorous results concerning the Lagrange multiplier theorem for the case involving equalities and inequalities. Theorem 8.1 Let U, Z1 , Z2 be Banach spaces. Consider a cone C in Z2 as above specified and such that if z1 ≤ θ and z2 < θ then z1 + z2 < θ , where z ≤ θ means that z ∈ −C and z < θ means that z ∈ (−C)◦ . The concerned order is supposed to be also that if z < θ , z∗ ≥ θ ∗ and z∗ ̸= θ then ⟨z, z∗ ⟩Z2 < 0. Furthermore, assume u0 ∈ U is a point of local minimum for F : U → R subject to G1 (u) = θ and G2 (u) ≤ θ , where G1 : U → Z1 , G2 : U → Z2 and F are Fr´echet differentiable at u0 ∈ U. Suppose also G′1 (u0 ) is onto and that there exist α > 0, K > 0 such that if ∥ϕ∥U < α then ∥G′1 (u0 + ϕ) − G′1 (u0 )∥ ≤ K∥ϕ∥U . Finally, suppose there exists ϕ0 ∈ U such that G′1 (u0 ) · ϕ0 = θ

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and G′2 (u0 ) · ϕ0 < θ . Under such hypotheses, there exists a Lagrange multiplier z∗0 = (z∗1 , z∗2 ) ∈ Z1∗ × Z2∗ such that F ′ (u0 ) + [G′1 (u0 )]∗ (z∗1 ) + [G′2 (u0 )]∗ (z∗2 ) = θ , z∗2 ≥ θ ∗ , and ⟨G2 (u0 ), z∗2 ⟩Z2 = 0. Proof 8.10

Let ϕ ∈ U be such that G′1 (u0 ) · ϕ = θ

and G′2 (u0 ) · ϕ = v − λ G2 (u0 ), for some v ≤ θ and λ ≥ 0. Select α ∈ (0, 1) and define ϕα = αϕ0 + (1 − α)ϕ. Observe that G1 (u0 ) = θ and G′1 (u0 ) · ϕα = θ so that as in the proof of the Lagrange multiplier theorem 9.2.1 we may find K1 > 0, ε > 0 and ψ0 (t) such that G1 (u0 + tϕα + t 2 ψ0 (t)) = θ , ∀|t| < ε, and ∥ψ0 (t)∥U < K1 , ∀|t| < ε. Observe that

= = = =

G′2 (u0 ) · ϕα αG′2 (u0 ) · ϕ0 + (1 − α)G′2 (u0 ) · ϕ αG′2 (u0 ) · ϕ0 + (1 − α)(v − λ G2 (u0 )) αG′2 (u0 ) · ϕ0 + (1 − α)v − (1 − α)λ G2 (u0 )) v0 − λ0 G2 (u0 ),

where, λ0 = (1 − α)λ , and v0 = αG′2 (u0 ) · ϕ0 + (1 − α)v < θ . Hence, for t > 0 G2 (u0 + tϕα + t 2 ψ0 (t)) = G2 (u0 ) + G′2 (u0 ) · (tϕα + t 2 ψ0 (t)) + r(t),

(8.51)

Constrained Variational Optimization

where lim

t→0+

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181

∥r(t)∥ = 0. t

Therefore from (9.12) we obtain G2 (u0 + tϕα + t 2 ψ0 (t)) = G2 (u0 ) + tv0 − tλ0 G2 (u0 ) + r1 (t), where

∥r1 (t)∥ = 0. t Observe that there exists ε1 > 0 such that if 0 < t < ε1 < ε, then lim

t→0+

v0 +

r1 (t) < θ, t

and G2 (u0 ) − tλ0 G2 (u0 ) = (1 − tλ0 )G2 (u0 ) ≤ θ . Hence G2 (u0 + tϕα + t 2 ψ0 (t)) < θ , if 0 < t < ε1 . From this there exists 0 < ε2 < ε1 such that F(u0 + tϕα + t 2 ψ0 (t)) − F(u0 ) = ⟨tϕα + t 2 ψ0 (t), F ′ (u0 )⟩U + r2 (t) ≥ 0,

(8.52)

where

|r2 (t)| = 0. t Dividing the last inequality by t > 0 we get lim

t→0+

⟨ϕα + tψ0 (t), F ′ (u0 )⟩U + r2 (t)/t ≥ 0, ∀0 < t < ε2 . Letting t → 0+ we obtain ⟨ϕα , F ′ (u0 )⟩U ≥ 0. Letting α → 0+ , we get

⟨ϕ, F ′ (u0 )⟩U ≥ 0,

if G′1 (u0 ) · ϕ = θ , and G′2 (u0 ) · ϕ = v − λ G2 (u0 ), for some v ≤ θ and λ ≥ 0. Define A = {(⟨ϕ, F ′ (u0 )⟩U + r, G′1 (u0 ) · ϕ, G′2 (u0 )ϕ − v + λ G(u0 )), ϕ ∈ U, r ≥ 0, v ≤ θ , λ ≥ 0}.

(8.53)

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Observe that A is a convex set with a non-empty interior. If G′1 (u0 ) · ϕ = θ , and G′2 (u0 ) · ϕ − v + λ G2 (u0 ) = θ , with v ≤ θ and λ ≥ 0 then

⟨ϕ, F ′ (u0 )⟩U ≥ 0,

so that ⟨ϕ, F ′ (u0 )⟩U + r ≥ 0. Moreover, if ⟨ϕ, F ′ (u0 )⟩ + r = 0, with r ≥ 0, G′1 (u0 ) · ϕ = θ , and G′2 (u0 ) · ϕ − v + λ G2 (u0 ) = θ , with v ≤ θ and λ ≥ 0, then we have ⟨ϕ, F ′ (u0 )⟩U ≥ 0, so that ⟨ϕ, F ′ (u0 )⟩U = 0, and r = 0. Hence (0, θ , θ ) is on the boundary of A. Therefore, by the Hahn-Banach theorem, geometric form, there exists (β , z∗1 , z∗2 ) ∈ R × Z1∗ × Z2∗ such that (β , z∗1 , z∗2 ) ̸= (0, θ , θ ) and β (⟨ϕ, F ′ (u0 )⟩U + r) + ⟨G′1 (u0 ) · ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 ) · ϕ − v + λ G2 (u0 ), z∗2 ⟩Z2 ≥ 0,

(8.54)

∀ ϕ ∈ U, r ≥ 0, v ≤ θ , λ ≥ 0. Suppose β = 0. Fixing all variable except v we get z∗2 ≥ θ . Thus, for ϕ = cϕ0 with arbitrary c ∈ R, v = θ , λ = 0, if z∗2 ̸= θ , then ⟨G′2 (u0 ) · ϕ0 , z∗2 ⟩Z2 < 0, so that we get z∗2 = θ . Since G′1 (u0 ) is onto, a similar reasoning lead us to z∗1 = θ , which contradicts (β , z∗1 , z∗2 ) ̸= (0, θ , θ ). Hence, β ̸= 0, and fixing all variables except r we obtain β > 0. There is no loss of generality in assuming β = 1. Again fixing all variables except v, we obtain z∗2 ≥ θ . Fixing all variables except λ , since G2 (u0 ) ≤ θ we get ⟨G2 (u0 ), z∗2 ⟩Z2 = 0.

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Finally, for r = 0, v = θ , λ = 0, we get ⟨ϕ, F ′ (u0 )⟩U + ⟨G′1 (u0 )ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 ) · ϕ, z∗2 ⟩Z2 ≥ 0, ∀ϕ ∈ U, that is, since obviously such an inequality is valid also for −ϕ, ∀ϕ ∈ U, we obtain ⟨ϕ, F ′ (u0 )⟩U + ⟨ϕ, [G′1 (u0 )]∗ (z∗1 )⟩U + ⟨ϕ, [G′2 (u0 )]∗ (z∗2 )⟩U = 0, ∀ϕ ∈ U, so that F ′ (u0 ) + [G′1 (u0 )]∗ (z∗1 ) + [G′2 (u0 )]∗ (z∗2 ) = θ . The proof is complete.

8.6

Second order necessary conditions

In this section, we establish second order necessary conditions for a class of constrained problems in Banach spaces. We highlight the next result is particularly applicable to optimization in Rn . Theorem 8.2 Let U, Z1 , Z2 be Banach spaces. Consider a cone C in Z2 as above specified and such that if z1 ≤ θ and z2 < θ then z1 + z2 < θ , where z ≤ θ means that z ∈ −C and z < θ means that z ∈ (−C)◦ . The concerned order is supposed to be also that if z < θ , z∗ ≥ θ ∗ and z∗ ̸= θ then ⟨z, z∗ ⟩Z2 < 0. Furthermore, assume u0 ∈ U is a point of local minimum for F : U → R subject to G1 (u) = θ and G2 (u0 ) ≤ θ , where G1 : U → Z1 , G2 : U → (Z2 )k and F are twice Fr´echet differentiable at u0 ∈ U. Assume G2 (u) = {(G2 )i (u)} where (G2 )i : U → Z2 , ∀i ∈ {1, ..., k} and define A = {i ∈ {1, ..., k} : (G2 )i (u0 ) = θ }, and also suppose that (G2 )i (u0 ) < θ , if i ̸∈ A. Moreover, suppose {G′1 (u0 ), {(G2 )′i (u0 )}i∈A } is onto and that there exist α > 0, K > 0 such that if ∥ϕ∥U < α then ∥G˜ ′ (u0 + ϕ) − G˜ ′ (u0 )∥ ≤ K∥ϕ∥U , where ˜ G(u) = {G1 (u), {(G2 )i (u)}i∈A }. Finally, suppose there exists ϕ0 ∈ U such that G′1 (u0 ) · ϕ0 = θ and G′2 (u0 ) · ϕ0 < θ .

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Under such hypotheses, there exists a Lagrange multiplier z∗0 = (z∗1 , z∗2 ) ∈ Z1∗ × (Z2∗ )k such that F ′ (u0 ) + [G′1 (u0 )]∗ (z∗1 ) + [G′2 (u0 )]∗ (z∗2 ) = θ , z∗2 ≥ (θ ∗ , ..., θ ∗ ) ≡ θk∗ , and ⟨(G2 )i (u0 ), (z∗2 )i ⟩Z = 0, ∀i ∈ {1, ..., k}, (z∗2 )i = θ ∗ , if i ̸∈ A, Moreover, defining L(u, z∗1 , z∗2 ) = F(u) + ⟨G1 (u), z∗1 ⟩Z1 + ⟨G2 (u), z∗2 ⟩Z2 , we have that 2 δuu L(u0 , z∗1 , z∗2 ; ϕ) ≥ 0, ∀ϕ ∈ V0 ,

where V0 = {ϕ ∈ U : G′1 (u0 ) · ϕ = θ , (G2 )′i (u0 ) · ϕ = θ , ∀i ∈ A}. Proof 8.11

Observe that A is defined by A = {i ∈ {1, ..., k} : (G2 )i (u0 ) = θ }.

Observe also that (G2 )i (u0 ) < θ , if i ̸∈ A. Hence the point u0 ∈ U is a local minimum for F(u) under the constraints G1 (u) = θ , and (G2 )i (u) ≤ θ , ∀i ∈ A. From the last Theorem 9.3.1 for such an optimization problem there exists a Lagrange multiplier (z∗1 , {(z∗2 )i∈A }) such that (z∗2 )i ≥ θ ∗ , ∀i ∈ A, and F ′ (u0 ) + [G′1 (u0 )]∗ (z∗1 ) + ∑ [(G2 )′i (u0 )]∗ ((z∗2 )i ) = θ .

(8.55)

i∈A

The choice (z∗2 )i = θ , if i ̸∈ A leads to the existence of a Lagrange multiplier (z∗1 , {(z∗2 )i∈A , (z∗2 )i̸∈A }) such that

(z∗1 , z∗2 ) =

z∗2 ≥ θk∗ and ⟨(G2 )i (u0 ), (z∗2 )i ⟩Z = 0, ∀i ∈ {1, ..., k}. Let ϕ ∈ V0 , that is, ϕ ∈ U, G′1 (u0 )ϕ = θ and (G2 )′i (u0 ) · ϕ = θ , ∀i ∈ A.

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˜ Recall that G(u) = {G1 (u), (G2 )i∈A (u)} and therefore, similarly as in the proof of the Lagrange multiplier theorem 9.2.1, we may obtain ψ0 (t), K > 0 and ε > 0 such that ˜ 0 + tϕ + t 2 ψ0 (t)) = θ , ∀|t| < ε, G(u and ∥ψ0 (t)∥ ≤ K, ∀|t| < ε. Also, if i ̸∈ A, we have that (G2 )i (u0 ) < θ , so that (G2 )i (u0 + tϕ + t 2 ψ0 (t)) = (G2 )i (u0 ) + G′i (u0 ) · (tϕ + t 2 ψ0 (t)) + r(t), where lim

t→0

∥r(t)∥ = 0, t

that is, (G2 )i (u0 + tϕ + t 2 ψ0 (t)) = (G2 )i (u0 ) + t(G2 )′i (u0 ) · ϕ + r1 (t), where, ∥r1 (t)∥ = 0, t and hence there exists, 0 < ε1 < ε, such that lim

t→0

(G2 )i (u0 + tϕ + t 2 ψ0 (t)) < θ , ∀|t| < ε1 < ε. Therefore, since u0 is a point of local minimum under the constraint G(u) ≤ θ , there exists 0 < ε2 < ε1 , such that F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) ≥ 0, ∀|t| < ε2 , so that, F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) = F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) +⟨G1 (u0 + tϕ + t 2 ψ0 (t)), z∗1 ⟩Z1 + ∑ ⟨(G2 )i (u0 + tϕ + t 2 ψ0 (t)), (z∗2 )i ⟩Z2 i∈A

−⟨G1 (u0 ), z∗1 ⟩Z1 − ∑ {⟨(G2 )i (u0 ), (z∗2 )i ⟩Z2 } i∈A

2

= F(u0 + tϕ + t ψ0 (t)) − F(u0 ) +⟨G1 (u0 + tϕ + t 2 ψ0 (t)), z∗1 ⟩Z1 − ⟨G1 (u0 ), z∗1 ⟩Z1 +⟨G2 (u0 + tϕ + t 2 ψ0 (t)), z∗2 ⟩Z2 − ⟨G2 (u0 ), z∗2 ⟩Z2 = L(u0 + tϕ + t 2 ψ0 (t)), z∗1 , z∗2 ) − L(u0 , z∗1 , z∗2 ) 1 2 L(u0 , z∗1 , z∗2 ;tϕ + t 2 ψ0 (t)) + r2 (t) = δu L(u0 , z∗1 , z∗2 ;tϕ + t 2 ψ0 (t)) + δuu 2 t2 2 = δ L(u0 , z∗1 , z∗2 ; ϕ + tψ0 (t)) + r2 (t) ≥ 0, ∀|t| < ε2 . 2 uu

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■

where lim |r2 (t)|/t 2 = 0.

t→0

To obtain the last inequality we have used δu L(u0 , z∗1 , z∗2 ;tϕ + t 2 ψ0 (t)) = 0 Dividing the last inequality by t 2 > 0 we obtain 1 2 δ L(u0 , z∗1 , z∗2 ; ϕ + tψ0 (t)) + r2 (t)/t 2 ≥ 0, ∀0 < |t| < ε2 , 2 uu and finally, letting t → 0 we get 1 2 δ L(u0 , z∗1 , z∗2 ; ϕ) ≥ 0. 2 uu The proof is complete.

8.7

On the Banach fixed point theorem

Now we recall a classical definition namely the Banach fixed theorem also known as the contraction mapping theorem. Definition 8.7.1 Let C be a subset of a Banach space U and let T : C → C be an operator. Thus T is said to be a contraction mapping if there exists 0 ≤ α < 1 such that ∥T (u) − T (v)∥U ≤ α∥u − v∥U , ∀u, v ∈ C. Remark 8.7.2 Observe that if ∥T ′ (u)∥U ≤ α < 1, on a convex set C then T is a contraction mapping, since by the mean value inequality, ∥T (u) − T (v)∥U ≤ sup{∥T ′ (u)∥}∥u − v∥U , ∀u, v ∈ C. u∈C

The next result is the base of our generalized method of lines. Theorem 8.7.3 (Contraction Mapping Theorem) Let C be a closed subset of a Banach space U. Assume T is contraction mapping on C, then there exists a unique u0 ∈ C such that u0 = T (u0 ). Moreover, for an arbitrary u1 ∈ C defining the sequence u2 = T (u1 ) and un+1 = T (un ), ∀n ∈ N we have un → u0 , in norm, as n → +∞. Proof 8.12

Let u1 ∈ C. Let {un } ⊂ C be defined by un+1 = T (un ), ∀n ∈ N.

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187

Hence, reasoning inductively ∥un+1 − un ∥U

= ≤ ≤ ≤ ≤

∥T (un ) − T (un−1 )∥U α∥un − un−1 ∥U α 2 ∥un−1 − un−2 ∥U ...... α n−1 ∥u2 − u1 ∥U , ∀n ∈ N.

(8.56)

Thus, for p ∈ N we have ∥un+p − un ∥U = ∥un+p − un+p−1 + un+p−1 − un+p−2 + ... − un+1 + un+1 − un ∥U ≤ ∥un+p − un+p−1 ∥U + ∥un+p−1 − un+p−2 ∥U + ... + ∥un+1 − un ∥U ≤ (α n+p−2 + α n+p−3 + ... + α n−1 )∥u2 − u1 ∥U ≤ α n−1 (α p−1 + α p−2 + ... + α 0 )∥u2 − u1 ∥U ! ≤ α n−1

∞

∑ αk

∥u2 − u1 ∥U

k=0

≤

α n−1 ∥u2 − u1 ∥U 1−α

(8.57)

Denoting n + p = m, we obtain ∥um − un ∥U ≤

α n−1 ∥u2 − u1 ∥U , ∀m > n ∈ N. 1−α

Let ε > 0. Since 0 ≤ α < 1 there exists n0 ∈ N such that if n > n0 then 0≤

α n−1 ∥u2 − u1 ∥U < ε, 1−α

so that ∥um − un ∥U < ε, if m > n > n0 . From this we may infer that {un } is a Cauchy sequence, and since U is a Banach space, there exists u0 ∈ U such that un → u0 , in norm, as n → ∞. Observe that ∥u0 − T (u0 )∥U

= ≤ ≤ →

∥u0 − un + un − T (u0 )∥U ∥u0 − un ∥U + ∥un − T (u0 )∥U ∥u0 − un ∥U + α∥un−1 − u0 ∥U 0, as n → ∞.

(8.58)

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Thus ∥u0 − T (u0 )∥U = 0. Finally, we prove the uniqueness. Suppose u0 , v0 ∈ C are such that u0 = T (u0 ) and v0 = T (v0 ). Hence, ∥u0 − v0 ∥U

= ∥T (u0 ) − T (v0 )∥U ≤ α∥u0 − v0 ∥U .

(8.59)

From this we get ∥u0 − v0 ||U ≤ 0, that is ∥u0 − v0 ∥U = 0. The proof is complete.

8.8 8.8.1

Sensitivity analysis Introduction

In this section, we state and prove the implicit function theorem for Banach spaces. A similar result may be found in Ito and Kunisch [51], page 31. We emphasize the result found in [51] is more general however, the proof present here is almost the same for a simpler situation. The general result found in [51] is originally from Robinson [61].

8.9

The implicit function theorem

Theorem 8.9.1 Let V,U,W be Banach spaces. Let F : V × U → W be a functions such that F(x0 , u0 ) = 0, where (x0 , u0 ) ∈ V ×U. Assume there exists r > 0 such that F is Fr´echet differentiable and Fx (x, u) is continuous in (x, u) in Br (x0 , u0 ). Suppose also [Fx (x0 , u0 )]−1 exists and is bounded so that there exists ρ > 0 such that 0 < ∥[Fx (x0 , u0 )]−1 ∥ ≤ ρ. Under such hypotheses, there exist 0 < ε1 < r/2 and 0 < ε2 < 1 such that for each u ∈ Bε1 (u0 ), there exists x ∈ Bε2 (x0 ) such that F(x, u) = 0, where we denote x = x(u) so that F(x(u), u) = 0.

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Moreover, there exists δ > 0 such that 0 < δ ρ < 1, such that for each u, v ∈ Bε1 (u0 ) we have ∥x(u) − x(v)∥ ≤

ρ 2δ ∥F(x(v), u) − F(x(v), v)∥. 1 − ρδ

Finally, if there exists K > 0 such that ∥Fu (x, u)∥ ≤ K, ∀(x, u) ∈ Bε2 (x0 ) × Bε1 (u0 ) so that ∥F(x, u) − F(x, v)∥ ≤ K∥u − v∥, ∀(x, u) ∈ Bε2 (x0 ) × Bε1 (u0 ), then ∥x(u) − x(v)∥ ≤ K1 ∥u − v∥, where K1 = K

Proof 8.13

ρ 2δ . 1−δρ

Let 0 < ε < r/2. Choose δ > 0 such that 0 < ρδ < 1.

Define T (x) = F(x0 , u0 ) + Fx (x0 , u0 )(x − x0 ) = Fx (x0 , u0 )(x − x0 ), and h(x, u) = F(x0 , u0 ) + Fx (x0 , u0 )(x − x0 ) − F(x, u). Choose 0 < ε3 < r/2 and 0 < ε2 < 1 such that Bε2 (u0 ) × Bε3 (x0 ) ⊂ Br (x0 , u0 ) and if (x, u) ∈ Bε2 (x0 ) × Bε3 (u0 ) then ∥Fx (x, u) − Fx (x0 , u0 )∥

0 be such that u + tϕ ∈ Bε1 (u0 ), ∀|t| < t0 . Observe that F(x(u + tϕ), u + tϕ) − F(x(u), u) = 0, ∀|t| < t0 . From the Fr´echet differentiability of F at (x(u), u), for 0 < |t| < t0 , we obtain Fx (x(u), u) · (x(u + tϕ) − x(u)) + Fu (x(u), u)(tϕ) +W (u, ϕ,t)(∥x(u + tϕ) − x(u)∥ + |t|∥ϕ∥) = 0,

(8.66)

Constrained Variational Optimization

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193

where W is such that W (u, ϕ,t) → 0, as t → 0, since x(u + tϕ) − x(u) → 0 and tϕ → 0, as t → 0. From this we obtain x(u + tϕ) − x(u) t

=

−[Fx (x(u), u)]−1 [[Fu (x(u), u)](ϕ) + r(u, ϕ,t)]

→

−[Fx (x(u), u)]−1 [Fu (x(u), u)](ϕ), as t → 0,

(8.67)

since ∥r(u, ϕ,t)∥

∥x(u + tϕ) − x(u)∥ + ∥ϕ∥ ≤ ∥W (u, ϕ,t)∥

t ≤ ∥W (u, ϕ,t)∥(K1 ∥ϕ∥ + ∥ϕ∥) → 0, as t → 0.

(8.68)

Summarizing, x(u + tϕ) − x(u) t = −[Fx (x(u), u)]−1 [Fu (x(u), u)](ϕ).

x′ (u, ϕ) =

lim

t→0

(8.69)

The proof is complete.

8.9.1

The main results about Gˆateaux differentiability

Again let V,U be Banach spaces and let F : V ×U → R be a functional. Fix u ∈ U and consider the problem of minimizing F(x, u) subject to G(x, u) ≤ θ and H(x, u) = θ . Here, the order and remaining details on the primal formulation are the same as those indicated in Section 8.4. Hence, for the specific case in which G : V ×U → [L p (Ω)]m1 and H : V ×U → [L p (Ω)]m2 , (the cases in which the co-domains of G and H are Rm1 and Rm2 respectively are dealt similarly) we redefine the concerned optimization problem, again for a fixed u ∈ U, by minimizing F(x, u) subject to {Gi (x, u) + v2i } = θ ,

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and H(x, u) = θ . ˜ u, v) = {Gi (x, u) + v2i } ≡ G(u) + v2 (from At this point we assume F(x, u), G(x, now on we use this general notation) and H(x, u) satisfy the hypotheses of the Lagrange multiplier theorem 9.2.1. Hence, for the fixed u ∈ U we assume there exists an optimal x ∈ V which locally minimizes F(x, u) under the mentioned constraints. From Theorem 9.2.1 there exist Lagrange multipliers λ1 , λ2 such that denoting [L p (Ω)]m1 and [L p (Ω)]m2 simply by L p , and defining ˜ u, λ1 , λ2 , v) = F(x, u) + ⟨λ1 , G(u) + v2 ⟩L p + ⟨λ2 , H(x, u)⟩L p , F(x, the following necessary conditions hold, F˜x (x, u) = Fx (x, u) + λ1 · Gx (x, u) + λ2 · Hx (x, u) = θ , 2

(8.70)

G(x, u) + v = θ ,

(8.71)

λ1 · v = θ ,

(8.72)

λ1 ≥ θ ,

(8.73)

H(x, u) = θ .

(8.74)

Clarifying the dependence on u, we denote the solution x, λ1 , λ2 , v by x(u), λ1 (u), λ2 (u), v(u), respectively. In particular, we assume that for a u0 ∈ U, x(u0 ), λ1 (u0 ), λ2 (u0 ), v(u0 ) satisfy the hypotheses of the implicit function theorem. Thus, for any u in an appropriate neighborhood of u0 , the corresponding x(u), λ1 (u), λ2 (u), v(u) are uniquely defined. We emphasize that from now on the main focus of our analysis is to evaluate variations of the optimal x(u), λ1 (u), λ2 (u), v(u) with variations of u in a neighborhood of u0 . For such an analysis, the main tool is the implicit function theorem and its main hypothesis is satisfied through the invertibility of the matrix of Fr´echet second derivatives. Hence, denoting, x0 = x(u0 ), (λ1 )0 = λ1 (u0 ), (λ2 )0 = λ2 (u0 ), v0 = v(u0 ), and A1 = Fx (x0 , u0 ) + (λ1 )0 · Gx (x0 , u0 ) + (λ2 )0 · Hx (x0 , u0 ), A2 = G(x0 , u0 ) + v20 A3 = H(x0 , u0 ), A4 = (λ1 )0 · v0 , we reiterate to assume that A1 = θ , A2 = θ , A3 = θ , A4 = θ ,

Constrained Variational Optimization

and M −1 to represent a bounded linear operator, where (A1 )x (A1 )λ1 (A1 )λ2 (A1 )v (A2 )x (A2 )λ (A2 )λ (A2 )v 1 2 M= (A3 )x (A3 )λ (A3 )λ (A3 )v 1 2 (A4 )x (A4 )λ1 (A4 )λ2 (A4 )v

■

195

where the derivatives are evaluated at (x0 , u0 , (λ1 )0 , (λ2 )0 , v0 ), so that, A Gx (x0 , u0 ) Hx (x0 , u0 ) θ Gx (x0 , u0 ) θ θ 2v0 M= Hx (x0 , u0 ) θ θ θ θ v0 θ (λ1 )0

(8.75)

(8.76)

where A = Fxx (x0 , u0 ) + (λ1 )0 · Gxx (x0 , u0 ) + (λ2 )0 · Hxx (x0 , u0 ). Moreover, also from the implicit function theorem, ∥(x(u), λ1 (u), λ2 (u), v(u)) − (x(u0 ), λ1 (u0 ), λ2 (u0 ), v(u0 ))∥ ≤ K∥u − u0 ∥,

(8.77)

for some appropriate K > 0, ∀u ∈ Br (u0 ), for some r > 0. We highlight to have denoted λ (u) = (λ1 (u), λ2 (u)). Let ϕ ∈ [C∞ (Ω)]k ∩U, where k depends on the vectorial expression of U. At this point we will be concerned with the following Gˆateaux variation evaluation ˜ δu F(x(u 0 ), u0 , λ (u0 ), v(u0 ); ϕ). Observe that ˜ δu F(x(u 0 ), u0 , λ (u0 ), v(u0 ); ϕ) = ˜ F(x(u0 + εϕ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) lim ε→0 ε ˜ F(x(u0 ), u0 , λ (u0 ), v(u0 )) − , ε so that ˜ δu F(x(u 0 ), u0 , λ (u0 ), v(u0 ); ϕ) = ˜ F(x(u 0 + εϕ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) lim ε→0 ε ˜ F(x(u ), u + εϕ, λ (u + εϕ), v(u0 + εϕ)) 0 0 0 − ε ˜ F(x(u 0 ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) + ε ˜ F(x(u ), u , λ (u ), v(u 0 0 0 0 )) − . ε

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However,

≤

˜ F(x(u 0 + εϕ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) ε ˜ F(x(u0 ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) − ε

F˜x (x(u0 + εϕ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) K∥ϕ∥

+K1 ∥x(u0 + εϕ) − x(u0 )∥ ≤ K1 K∥ϕ∥ε → 0, as ε → 0. In these last inequalities we have used

x(u0 + εϕ) − x(u0 )

≤ K∥ϕ∥, lim sup

ε ε→0 and F˜x (x(u0 + εϕ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) = θ . On the other hand, ˜ F(x(u 0 ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) ε ˜ F(x(u0 ), u0 , λ (u0 ), v(u0 )) − ε ˜ F(x(u0 ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) = ε ˜ F(x(u0 ), u0 + εϕ, λ (u0 ), v(u0 )) − ε

˜ F(x(u 0 ), u0 + εϕ, λ (u0 ), v(u0 )) ε ˜ F(x(u 0 ), u0 , λ (u0 ), v(u0 )) − ε

+

Now observe that ˜ F(x(u 0 ), u0 + εϕ, λ (u0 + εϕ), v(u0 + εϕ)) ε ˜ F(x(u0 ), u0 + εϕ, λ (u0 ), v(u0 )) − ε ⟨λ1 (u0 + εϕ), G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ⟩L p = ε ⟨λ1 (u0 ), G(x(u0 ), u0 + εϕ) + v(u0 )2 ⟩L p − ε ⟨λ2 (u0 + εϕ) − λ2 (u0 ), H(x(u0 ), u0 + εϕ)⟩L p + . ε

(8.78)

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Also,

≤

+

⟨λ1 (u0 + εϕ), G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ⟩L p ε ⟨λ1 (u0 ), G(x(u0 ), u0 + εϕ) + v(u0 )2 ⟩L p − ε ⟨λ1 (u0 + εϕ), G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ⟩L p ε ⟨λ1 (u0 ), G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ⟩L p − ε ⟨λ1 (u0 ), G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ⟩L p ε ⟨λ1 (u0 ), G(x(u0 ), u0 + εϕ) + v(u0 )2 ⟩L p − ε

K∥ϕ∥ ∥G(x(u0 ), u0 + εϕ) + v(u0 + εϕ)2 ∥ ε K∥ϕ∥ε + ∥λ1 (u0 )(v(u0 + εϕ) + v(u0 ))∥ ε → 0 as ε → 0. ≤

ε

To obtain the last inequalities we have used

λ1 (u0 + εϕ) − λ1 (u0 )

≤ K∥ϕ∥, lim sup

ε ε→0 λ1 (u0 )v(u0 ) = θ , λ1 (u0 )v(u0 + εϕ) → θ , as ε → 0, and

λ1 (u0 )(v(u0 + εϕ)2 − v(u0 )2 )

ε

λ1 (u0 )(v(u0 + εϕ) + v(u0 ))(v(u0 + εϕ) − v(u0 ))

=

ε ∥λ1 (u0 )(v(u0 + εϕ) + v(u0 ))∥K∥ϕ∥ε ≤ ε → 0, as ε → 0.

(8.79)

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Finally,

≤ →

⟨λ2 (u0 + εϕ) − λ2 (u0 ), H(x(u0 ), u0 + εϕ)⟩L p ε Kε∥ϕ∥ ∥H(x(u0 ), u0 + εϕ)∥ ε 0, as ε → 0.

To obtain the last inequalities we have used

λ2 (u0 + εϕ) − λ2 (u0 )

≤ K∥ϕ∥,

lim sup

ε ε→0 and H(x(u0 ), u0 + εϕ) → θ , as ε → 0. From these last results, we get ˜ δu F(x(u 0 ), u0 , λ (u0 ), v(u0 ); ϕ) = ˜ F(x(u 0 ), u0 + εϕ, λ (u0 ), v(u0 )) ε→0 ε ˜ F(x(u0 ), u0 , λ (u0 ), v(u0 )) − ε

lim

= ⟨Fu (x(u0 ), u0 ), ϕ⟩U + ⟨λ1 (u0 ) · Gu (x(u0 ), u0 ), ϕ⟩L p +⟨λ2 (u0 ) · Hu (x(u0 ), u0 ), ϕ⟩L p . In the last lines we have proven the following corollary of the implicit function theorem, Corollary 8.1 Suppose (x0 , u0 , (λ1 )0 , (λ2 )0 , v0 ) is a solution of the system (8.70), (8.71),(8.72), (8.74), and assume the corresponding hypotheses of the implicit function theorem ˜ u, λ1 , λ2 , v) is such that the Fr´echet second derivative are satisfied. Also assume F(x, F˜xx (x, u, λ1 , λ2 ) is continuous in a neighborhood of (x0 , u0 , (λ1 )0 , (λ2 )0 ).

Constrained Variational Optimization

Under such hypotheses, for a given ϕ ∈ [C∞ (Ω)]k , denoting ˜ F1 (u) = F(x(u), u, λ1 (u), λ2 (u), v(u)), we have δ (F1 (u); ϕ)|u=u0 = ⟨Fu (x(u0 ), u0 ), ϕ⟩U + ⟨λ1 (u0 ) · Gu (x(u0 ), u0 ), ϕ⟩L p +⟨λ2 (u0 ) · Hu (x(u0 ), u0 ), ϕ⟩L p .

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Chapter 9

On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces

9.1

Introduction

This article develops optimality conditions for a large class of non-smooth variational models. The main results are based on standard tools of functional analysis and calculus of variations. Firstly, we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach space context. We highlight the results which in general are well known, however, some novelties related to the proof procedures are introduced, which are in general softer than those concerning the present literature. Remark 9.1.1 This chapter has been accepted for publication in a similar article format by the Universal Wiser Journal Contemporary Mathematics, reference [19]. ©2023 Fabio Silva Botelho, On Lagrange Multiplier Theorems for Non-Smooth Optimization for a Large Class of Variational Models in Banach Spaces.

On Lagrange Multiplier Theorems for Non-Smooth Optimization

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201

DOI: https://doi.org/10.37256/cm.3420221711 This is an open-access article distributed under a CC BY license (Creative Commons Attribution 4.0 International License) https://creativecommons.org/licenses/by/4.0/. The main references for this article are [79, 12, 13]. Indeed, the results presented here are, in some sense, extensions of the previous ones found in F. Clarke [79]. We also highlight specific details on the function spaces addressed, and concerning functional analysis and Lagrange multiplier basic results may be found in [1, 77, 26, 24, 79, 78, 12, 13]. Related subjects are addressed in [80, 76]. Specifically in [80], the authors propose an augmented Lagrangian method for the solution of constrained optimization problems suitable for a large class of variational models. At this point, we highlight that the main novelties mentioned in the abstract are specified in the first three paragraphs of Section 12, and are applied in the statements and proofs of Theorems 9.2.1 and 9.3.1. Finally, fundamental results on the calculus of variations are addressed in [82]. We start with some preliminary results and basic definitions. The first result we present is the Hahn-Banach Theorem in its analytic form. Concerning our context, we have assumed the hypothesis that the space U is a Banach space but indeed such a result is much more general. Theorem 9.1.2 (The Hahn-Banach theorem) Let U be a Banach space. Consider a functional p : U → R such that p(λ u) = λ p(u), ∀u ∈ U, λ > 0,

(9.1)

p(u + v) ≤ p(u) + p(v), ∀u, v ∈ U.

(9.2)

and

Let V ⊂ U be a proper subspace of U and let g : V → R be a linear functional such that g(u) ≤ p(u), ∀u ∈ V.

(9.3)

Under such hypotheses, there exists a linear functional f : U → R such that g(u) = f (u), ∀u ∈ V,

(9.4)

f (u) ≤ p(u), ∀u ∈ U.

(9.5)

and

For a proof, please see [26, 12, 13]. Here we introduce the definition of topological dual space. Definition 9.1.3 (Topological dual spaces) Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined

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on U. We suppose such a dual space of U, may be represented by another Banach space U ∗ , through a bilinear form ⟨·, ·⟩U : U ×U ∗ → R (here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given f : U → R linear and continuous, we assume the existence of a unique u∗ ∈ U ∗ such that f (u) = ⟨u, u∗ ⟩U , ∀u ∈ U.

(9.6)

The norm of f , denoted by ∥ f ∥U ∗ , is defined as ∥ f ∥U ∗ = sup{|⟨u, u∗ ⟩U | : ∥u∥U ≤ 1} ≡ ∥u∗ ∥U ∗ .

(9.7)

u∈U

At this point we present the Hahn-Banach Theorem in its geometric form. Theorem 9.1.4 (The Hahn-Banach theorem, the geometric form) Let U be a Banach space and let A, B ⊂ U be two non-empty, convex sets such that A ∩ B = 0/ and A is open. Under such hypotheses, there exists a closed hyperplane which separates A and B, that is, there exist α ∈ R and u∗ ∈ U ∗ such that u∗ ̸= 0 and ⟨u, u∗ ⟩U ≤ α ≤ ⟨v, u∗ ⟩U , ∀u ∈ A, v ∈ B. For a proof, please see [26, 12, 13]. Another important definition, is the one concerning locally Lipschitz functionals. Definition 9.1.5 Let U be a Banach space and let F : U → R be a functional. We say that F is locally Lipschitz at u0 ∈ U if there exist r > 0 and K > 0 such that |F(u) − F(v)| ≤ K∥u − v∥U , ∀u, v ∈ Br (u0 ). In this definition, we have denoted Br (u0 ) = {v ∈ U : ∥u0 − v∥U < r}. The next definition is established as those found in the reference [79]. More specifically, such a next one, corresponds to the definition of generalized directional derivative found in Section 10.1 at page 194, in reference [79]. Definition 9.1.6 Let U be a Banach space and let F : U → R be a locally Lipschitz functional at u ∈ U. Let ϕ ∈ U. Under such statements, we define F(un + tn ϕ) − F(un ) + : un → u in U, tn → 0 . Hu (ϕ) = sup lim sup tn n→∞ ({un },{tn })⊂U×R+ We also define the generalized local sub-gradient set of F at u, denoted by ∂ 0 F(u), by ∂ 0 F(u) = {u∗ ∈ U ∗ : ⟨ϕ, u∗ ⟩U ≤ Hu (ϕ), ∀ϕ ∈ U}.

On Lagrange Multiplier Theorems for Non-Smooth Optimization

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■

We also highlight such a last definition of generalized local sub-gradient is similar as the definition of generalized gradient, which may be found in Section 10.13, at page 196, in the book [79]. In the next lines we prove some relevant auxiliary results. Proposition 9.1.7 Considering the context of the last two definitions, we have 1. Hu (ϕ1 + ϕ2 ) ≤ Hu (ϕ1 ) + Hu (ϕ2 ), ∀ϕ1 , ϕ2 ∈ U. 2. Hu (λ ϕ) = λ Hu (ϕ), ∀λ > 0, ϕ ∈ U. Proof 9.1 Let ϕ1 , ϕ2 ∈ U. Observe that Hu (ϕ1 + ϕ2 ) =

sup ({un },{tn })⊂U×R+

F(un + tn (ϕ1 + ϕ2 )) − F(un ) : un → u in U, tn → 0+ lim sup tn n→∞

=

sup ({un },{tn

F(un + tn (ϕ1 + ϕ2 ) − F(uu + tn ϕ2 ) + F(un + tn ϕ2 )) − F(un ) tn +

lim sup

})⊂U×R+

n→∞

: un → u in U, tn → 0 F(vn + tn ϕ1 ) − F(vn ) sup : vn → u in U, tn → 0+ lim sup tn n→∞ ({vn },{tn })⊂U×R+ F(un + tn ϕ2 ) − F(un ) + + sup : un → u in U, tn → 0 lim sup tn n→∞ ({un },{tn })⊂U×R+

≤

=

Hu (ϕ1 ) + Hu (ϕ2 ).

(9.8)

Let ϕ ∈ U and λ > 0. Thus, Hu (λ ϕ) =

sup ({un },{tn })⊂U×R+

lim sup n→∞

F(un + tn (λ ϕ)) − F(un ) : un → u in U, tn → 0+ tn

F(un + tn (λ ϕ)) − F(un ) + λ sup : un → u in U, tn → 0 lim sup λtn n→∞ ({un },{tn })⊂U×R+ F(un + tˆn (ϕ)) − F(un ) λ sup : un → u in U, tˆn → 0+ lim sup tˆn n→∞ ({un },{tˆn })⊂U×R+

= = =

λ Hu (ϕ).

The proof is complete.

(9.9)

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9.2

The Lagrange multiplier theorem for equality constraints and non-smooth optimization

In this section, we state and prove a Lagrange multiplier theorem for non-smooth optimization. This first one is related to equality constraints. Here we refer to a related result in the Theorem 10.45 at page 220, in the book [79]. We emphasize that in such a result, in this mentioned book, the author assumes the function which defines the constraints to be continuously differentiable in a neighborhood of the point in question. In our next result, we do not assume such a hypothesis. Indeed, our hypotheses are different and in some sense weaker. More specifically, we assume the continuity of the Frech´et derivative G′ (u) of a concerning constraint G(u) only at the optimal point u0 and not necessarily in a neighborhood, as properly indicated in the next lines. Theorem 9.2.1 Let U and Z be Banach spaces. Assume u0 is a local minimum of F(u) subject to G(u) = θ , where F : U → R is locally Lipschitz at u0 and G : U → Z is a Fr´echet differentiable transformation such that G′ (u0 ) maps U onto Z. Finally, assume there exist α > 0 and K > 0 such that if ∥ϕ∥U < α then, ∥G′ (u0 + ϕ) − G′ (u0 )∥ ≤ K∥ϕ∥U . Under such assumptions, there exists z∗0 ∈ Z ∗ such that θ ∈ ∂ 0 F(u0 ) + (G′ (u0 )∗ )(z∗0 ), that is, there exist u∗ ∈ ∂ 0 F(u0 ) and z∗0 ∈ Z ∗ such that u∗ + [G′ (u0 )]∗ (z∗0 ) = θ , so that, ⟨ϕ, u∗ ⟩U + ⟨G′ (u0 )ϕ, z∗0 ⟩Z = 0, ∀ϕ ∈ U. Proof 9.2

Let ϕ ∈ U be such that G′ (u0 )ϕ = θ .

From the proof of Theorem 11.3.2 at page 292, in [12], there exist ε0 > 0, K1 > 0 and {ψ0 (t), 0 < |t| < ε0 } ⊂ U such that ∥ψ0 (t)∥U ≤ K1 , ∀0 < |t| < ε0 , and G(u0 + tϕ + t 2 ψ0 (t)) = θ , ∀0 < |t| < ε0 .

On Lagrange Multiplier Theorems for Non-Smooth Optimization

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205

From this and the hypotheses on u0 , there exists 0 < ε1 < ε0 such that F(u0 + tϕ + t 2 ψ0 (t)) ≥ F(u0 ), ∀0 < |t| < ε1 , so that

F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) ≥ 0, ∀0 < t < ε1 . t

Hence, F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) t F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 + t 2 ψ0 (t)) + F(u0 + t 2 ψ0 (t)) − F(u0 ) t F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 + t 2 ψ0 (t)) + Kt∥ψ0 (t)∥U , ∀0 < t < min{r, ε1 }. (9.10) t

0≤ = ≤

From this, we obtain 0 ≤ lim sup t→0+

= lim sup t→0+

≤ lim sup t→0+

= lim sup t→0+

F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 ) t

F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 + t 2 ψ0 (t)) + F(u0 + t 2 ψ0 (t)) − F(u0 ) t F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 + t 2 ψ0 (t)) + lim sup Kt∥ψ0 (t)∥U t t→0+ F(u0 + tϕ + t 2 ψ0 (t)) − F(u0 + t 2 ψ0 (t)) t

≤ Hu0 (ϕ).

(9.11)

Summarizing, Hu0 (ϕ) ≥ 0, ∀ϕ ∈ N(G′ (u0 )). Hence, Hu0 (ϕ) ≥ 0 = ⟨ϕ, θ ⟩U , ∀ϕ ∈ N(G′ (u0 )). From the Hahn-Banach Theorem, the functional f ≡0 defined on N(G′ (u0 )) may be extended to U through a linear functional f1 : U → R such that f1 (ϕ) = 0, ∀ϕ ∈ N[G′ (u0 )] and f1 (ϕ) ≤ Hu0 (ϕ), ∀ϕ ∈ U. Since from the local Lipschitz property Hu0 is bounded, so is f1 .

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Therefore, there exists u∗ ∈ U ∗ such that f1 (ϕ) = ⟨ϕ, u∗ ⟩U ≤ Hu0 (ϕ), ∀ϕ ∈ U, so that u∗ ∈ ∂ 0 F(u0 ). Finally, observe that ⟨ϕ, u∗ ⟩U = 0, ∀ϕ ∈ N(G′ (u0 )). Since G′ (u0 ) is onto (closed range), from a well known result for linear operators, we have that u∗ ∈ R[G′ (u0 )∗ ]. Thus, there exists, z∗0 ∈ Z ∗ such that u∗ = [G′ (u0 )∗ ](−z∗0 ), so that u∗ + [G′ (u0 )∗ ](z∗0 ) = θ . From this, we obtain ⟨ϕ, u∗ ⟩U + ⟨ϕ, [G′ (u0 )∗ ](z∗0 )⟩U = 0, that is, ⟨ϕ, u∗ ⟩U + ⟨G′ (u0 )ϕ, (z∗0 )⟩Z = 0, ∀ϕ ∈ U The proof is complete.

9.3

The Lagrange multiplier theorem for equality and inequality constraints for non-smooth optimization

In this section, we develop a rigorous result concerning the Lagrange multiplier theorem for the case involving equalities and inequalities. Theorem 9.3.1 Let U, Z1 , Z2 be Banach spaces. Consider a cone C in Z2 (as specified at Theorem 11.1 in [12]) such that if z1 ≤ θ and z2 < θ then z1 + z2 < θ , where z ≤ θ means that z ∈ −C and z < θ means that z ∈ (−C)◦ . The concerned order is supposed to be also that if z < θ , z∗ ≥ θ ∗ and z∗ ̸= θ then ⟨z, z∗ ⟩Z2 < 0. Furthermore, assume u0 ∈ U is a point of local minimum for F : U → R subject to G1 (u) = θ and G2 (u) ≤ θ , where G1 : U → Z1 , G2 : U → Z2 are Fr´echet differentiable transformations and F locally Lipschitz at u0 ∈ U. Suppose also G′1 (u0 ) is onto and that there exist α > 0, K > 0 such that if ∥ϕ∥U < α then ∥G′1 (u0 + ϕ) − G′1 (u0 )∥ ≤ K∥ϕ∥U .

On Lagrange Multiplier Theorems for Non-Smooth Optimization

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207

Finally, suppose there exists ϕ0 ∈ U such that G′1 (u0 ) · ϕ0 = θ and G′2 (u0 ) · ϕ0 < θ . Under such hypotheses, there exists a Lagrange multiplier z∗0 = (z∗1 , z∗2 ) ∈ Z1∗ × Z2∗ such that θ ∈ ∂ 0 F(u0 ) + [G′1 (u0 )∗ ](z∗1 ) + [G′2 (u0 )∗ ](z∗2 ), z∗2 ≥ θ ∗ , and ⟨G2 (u0 ), z∗2 ⟩Z2 = 0, that is, there exists u∗ ∈ ∂ 0 F(u0 ) and a Lagrange multiplier z∗0 = (z∗1 , z∗2 ) ∈ Z1∗ × Z2∗ such that u∗ + [G′1 (u0 )]∗ (z∗1 ) + [G′2 (u0 )]∗ (z∗2 ) = θ , so that ⟨ϕ, u∗ ⟩U + ⟨ϕ, G′1 (u0 )∗ (z∗1 )⟩U + ⟨ϕ, G′2 (u0 )∗ (z∗2 )⟩U = 0, that is, ⟨ϕ, u∗ ⟩U + ⟨G′1 (u0 )ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 )ϕ, z∗2 ⟩Z2 = 0, ∀ϕ ∈ U. Proof 9.3

Let ϕ ∈ U be such that G′1 (u0 ) · ϕ = θ

and G′2 (u0 ) · ϕ = v − λ G2 (u0 ), for some v ≤ θ and λ ≥ 0. For α ∈ (0, 1) define ϕα = αϕ0 + (1 − α)ϕ. Observe that G1 (u0 ) = θ and G′1 (u0 ) · ϕα = θ so that as in the proof of the Lagrange multiplier Theorem 11.3.2 in [12], we may find K1 > 0, ε > 0 and ψ0α (t) such that G1 (u0 + tϕα + t 2 ψ0α (t)) = θ , ∀|t| < ε, ∀α ∈ (0, 1) and ∥ψ0α (t)∥U < K1 , ∀|t| < ε, ∀α ∈ (0, 1). Observe that

= = = =

G′2 (u0 ) · ϕα αG′2 (u0 ) · ϕ0 + (1 − α)G′2 (u0 ) · ϕ αG′2 (u0 ) · ϕ0 + (1 − α)(v − λ G2 (u0 )) αG′2 (u0 ) · ϕ0 + (1 − α)v − (1 − α)λ G2 (u0 )) v0 − λ0 G2 (u0 ),

(9.12)

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where, λ0 = (1 − α)λ , and v0 = αG′2 (u0 ) · ϕ0 + (1 − α)v < θ . Hence, for t > 0 G2 (u0 + tϕα + t 2 ψ0α (t)) = G2 (u0 ) + G′2 (u0 ) · (tϕα + t 2 ψ0α (t)) + r(t), where

∥r(t)∥ = 0. t

lim

t→0+

Therefore from (9.12) we obtain G2 (u0 + tϕα + t 2 ψ0α (t)) = G2 (u0 ) + tv0 − tλ0 G2 (u0 ) + r1 (t), where lim

t→0+

∥r1 (t)∥ = 0. t

Observe that there exists ε1 > 0 such that if 0 < t < ε1 < ε, then v0 +

r1 (t) < θ, t

and G2 (u0 ) − tλ0 G2 (u0 ) = (1 − tλ0 )G2 (u0 ) ≤ θ . Hence G2 (u0 + tϕα + t 2 ψ0α (t)) < θ , if 0 < t < ε1 . From this there exists 0 < ε2 < ε1 such that F(u0 + tϕα + t 2 ψ0α (t)) ≥ F(u0 ), ∀0 < t < ε2 , α ∈ (0, 1). In particular F(u0 + tϕt + t 2 ψ0t (t)) ≥ F(u0 ), ∀0 < t < min{1, ε2 }, so that

F(u0 + tϕt + t 2 ψ0t (t)) − F(u0 ) ≥ 0, ∀0 < t < min{1, ε2 }, t

that is, F(u0 + tϕ + t 2 (ψ0t (t) + ϕ0 − ϕ)) − F(u0 ) ≥ 0, ∀0 < t < min{1, ε2 }. t

On Lagrange Multiplier Theorems for Non-Smooth Optimization

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209

From this we obtain, 0 ≤ lim sup t→0+

F(u0 + tϕ + t 2 (ψ0t (t) + ϕ0 − ϕ)) − F(u0 ) t

F(u0 + tϕ + t 2 (ψ0t (t) + ϕ0 − ϕ)) − F(u0 + t 2 (ψ0t (t) + ϕ0 − ϕ)) t t→0+ 2 t F(u0 + t (ψ0 (t) + ϕ0 − ϕ)) − F(u0 ) + t

= lim sup

≤ lim sup t→0+

F(u0 + tϕ + t 2 (ψ0t (t) + ϕ0 − ϕ)) − F(u0 + t 2 (ψ0t (t) + ϕ0 − ϕ)) t

+ lim sup Kt∥ψ0t (t) + ϕ0 − ϕ∥U t→0+

= lim sup t→0+

F(u0 + tϕ + t 2 (ψ0t (t) + ϕ0 − ϕ)) − F(u0 + t 2 (ψ0t (t) + ϕ0 − ϕ)) t

≤ Hu0 (ϕ).

(9.13)

Summarizing, we have Hu0 (ϕ) ≥ 0, if G′1 (u0 ) · ϕ = θ , and G′2 (u0 ) · ϕ = v − λ G2 (u0 ), for some v ≤ θ and λ ≥ 0. Define A = {Hu0 (ϕ) + r, G′1 (u0 ) · ϕ, G′2 (u0 )ϕ − v + λ G2 (u0 )), ϕ ∈ U, r ≥ 0, v ≤ θ , λ ≥ 0}.

(9.14)

From the convexity of Hu0 (and the hypotheses on G′1 (u0 ) and G′2 (u0 )) we have that A is a convex set (with a non-empty interior). If G′1 (u0 ) · ϕ = θ , and G′2 (u0 ) · ϕ − v + λ G2 (u0 ) = θ , with v ≤ θ and λ ≥ 0 then Hu0 (ϕ) ≥ 0, so that Hu0 (ϕ) + r ≥ 0, ∀r ≥ 0. From this and Hu0 (θ ) = 0,

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we have that (0, θ , θ ) is on the boundary of A. Therefore, by the Hahn-Banach theorem, geometric form, there exists (β , z∗1 , z∗2 ) ∈ R × Z1∗ × Z2∗ such that (β , z∗1 , z∗2 ) ̸= (0, θ , θ ) and β (Hu0 (ϕ) + r) + ⟨G′1 (u0 ) · ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 ) · ϕ − v + λ G2 (u0 ), z∗2 ⟩Z2 ≥ 0,

(9.15)

∀ ϕ ∈ U, r ≥ 0, v ≤ θ , λ ≥ 0. Suppose β = 0. Fixing all variable except v we get z∗2 ≥ θ . Thus, for ϕ = cϕ0 with arbitrary c ∈ R, v = θ , λ = 0, if z∗2 ̸= θ , then ⟨G′2 (u0 ) · ϕ0 , z∗2 ⟩Z2 < 0 so that, letting c → +∞, we get a contradiction through (9.15), so that z∗2 = θ . Since G′1 (u0 ) is onto, a similar reasoning lead us to z∗1 = θ , which contradicts (β , z∗1 , z∗2 ) = ̸ (0, θ , θ ). Hence, β ̸= 0, and fixing all variables except r we obtain β > 0. There is no loss of generality in assuming β = 1. Again fixing all variables except v, we obtain z∗2 ≥ θ . Fixing all variables except λ , since G2 (u0 ) ≤ θ we obtain ⟨G2 (u0 ), z∗2 ⟩Z2 = 0. Finally, for r = 0, v = θ , λ = 0, we get Hu0 (ϕ) + ⟨G′1 (u0 )ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 ) · ϕ, z∗2 ⟩Z2 ≥ 0 = ⟨ϕ, θ ⟩U , ∀ϕ ∈ U. From this, θ ∈ ∂ 0 (F(u0 ) + ⟨G1 (u0 ), z∗1 ⟩Z1 + ⟨G2 (u0 ), z∗2 ⟩Z2 ) = ∂ 0 F(u0 ) + [G′1 (u0 )∗ ](z∗1 ) + [G′2 (u0 )∗ ](z∗2 ),

so that there exists u∗ ∈ ∂ 0 F(u0 ), such that u∗ + [G′1 (u0 )∗ ](z∗1 ) + [G′2 (u0 )∗ ](z∗2 ) = θ , so that ⟨ϕ, u∗ ⟩U + ⟨ϕ, G′1 (u0 )∗ (z∗1 )⟩U + ⟨ϕ, G′2 (u0 )∗ (z∗2 )⟩U = 0, that is, ⟨ϕ, u∗ ⟩U + ⟨G′1 (u0 )ϕ, z∗1 ⟩Z1 + ⟨G′2 (u0 )ϕ, z∗2 ⟩Z2 = 0, ∀ϕ ∈ U. The proof is complete.

9.4

Conclusion

In this article, we have presented an approach on Lagrange multiplier theorems for non-smooth variational optimization in a general Banach space context. The results are based on standard tools of functional analysis, calculus of variations and optimization. We emphasize, in the present article, no hypotheses concerning convexity are assumed and the results indeed are valid for such a more general Banach space context.

DUALITY PRINCIPLES AND RELATED NUMERICAL EXAMPLES THROUGH THE GENERALIZED METHOD OF LINES

III

Chapter 10

A Convex Dual Formulation for a Large Class of Non-Convex Models in Variational Optimization

10.1

Introduction

This short communication develops a convex dual variational formulation for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis, and the duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. Such results are based on the works of J.J. Telega and W.R. Bielski [10, 11, 74, 75], and on a D.C. optimization approach developed in Toland [81]. At this point, we start to describe the primal and dual variational formulations. Let Ω ⊂ R3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω.

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213

For the primal formulation we consider the functional J : U → R where γ 2

J(u) =

+

Z

∇u · ∇u dx

Ω

α 2

Z Ω

(u2 − β )2 dx − ⟨u, f ⟩L2 .

(10.1)

Here we assume α > 0, β > 0, γ > 0, U = W01,2 (Ω), f ∈ L2 (Ω). Moreover we denote Y = Y ∗ = L2 (Ω). Define also G : U → R by G(u) =

α 2

Z

(u2 − β )2 dx +

Ω

K 2

Z

u2 dx,

Ω

and F : U → R by γ K ∇u · ∇u dx + 2 Ω 2 It is worth highlighting that in such a case F(u) = −

Z

Z Ω

u2 dx + ⟨u, f ⟩L2 ,

J(u) = −F(u) + G(u), ∀u ∈ U. From now and on, we assume a finite dimensional version for this model, in a finite elements of finite differences context, where, for not relabeled operators and spaces, we also assume, γ∇2 + K > 0 in an appropriate matrices sense. Furthermore, define A+ = {u ∈ U : δ 2 J(u) ≥ 0} (A+ )0 = {u ∈ U : δ 2 J(u) > 0}, C+ = {u ∈ U : u f ≥ 0, in Ω}, E + = A+ ∩C+ and the following specific polar functionals specified, namely, G∗ : Y ∗ → R by G∗ (v∗1 ) =

sup {⟨u, v∗1 ⟩L2 − G(u)}

(10.2)

u∈A+

and F ∗ : Y ∗ → R by F ∗ (v∗1 ) = =

sup {⟨u, v∗1 ⟩L2 − F(u)}

u∈U

1 2

(v∗1 − f )2 dx. 2 Ω γ∇ + K

Z

(10.3)

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Define also J ∗ : Y ∗ → R by J ∗ (v∗1 ) = F ∗ (v∗1 ) − G∗ (v∗1 ) Observe that there exists a Lagrange multiplier λ ∈ W01,2 (Ω) such that Z γ ∇λ · ∇λ dx G∗ (v∗1 ) = sup ⟨u, v∗1 ⟩L2 − G(u) + 2 Ω u∈U Z Z 6α λ 2 u2 dx − αβ λ 2 dx . + 2 Ω Ω

(10.4)

Define now G2 : Y ∗ ×U ×U → R by G2 (v∗1 , u, λ ) = ⟨u, v∗1 ⟩L2 − G(u) +

γ 2

Z

∇λ · ∇λ dx +

Ω

6α 2

Z Ω

λ 2 u2 dx − αβ

Z

λ 2 dx.

Ω

Observe also that G∗ (v∗1 ) = G2 (v∗1 , u, ˆ λˆ ), where uˆ = u(v∗1 ) and λˆ = λ (v∗1 ) are such that ∂ G2 (v∗1 , u, ˆ λˆ ) = 0, ∂u and

∂ G2 (v∗1 , u, ˆ λˆ ) = 0. ∂λ On the other hand, ˆ λˆ ) ∂ 2 G2 (v∗1 , u, ˆ λˆ ) ∂ uˆ ∂ 2 G2 (v∗1 , u, ˆ λˆ ) ∂ λˆ ∂ 2 G∗ (v∗1 ) ∂ 2 G2 (v∗1 , u, = + + + . ∗ ∗ ∗ ∗ ∗ ∂ (v1 )2 ∂ (v1 )2 ∂ v1 ∂ u ∂ v1 ∂ v1 ∂ λ ∂ v∗1 Moreover, ∂ 2 G2 (v∗1 , u, ˆ λˆ ) = 0, ∂ (v∗1 )2 ∂ 2 G2 (v∗1 , u, ˆ λˆ ) = 1, ∂ v∗1 ∂ u

and

∂ 2 G2 (v∗1 , u, ˆ λˆ ) = 0. ∗ ∂ v1 ∂ λ From these last results we get ∂ 2 G∗ (v∗1 ) ∂ uˆ = ∗. ∗ 2 ∂ (v1 ) ∂ v1

A Convex Dual Formulations for Non-Convex Models

However from

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215

∂ G2 (v∗1 , u, ˆ λˆ ) = 0, ∂u

we have v∗1 − 2α(uˆ2 − β )uˆ − K uˆ + 6α λˆ 2 uˆ = 0 Taking the variation in v∗1 in this last equation, we obtain 1 − 6α uˆ2 −K

∂ uˆ ∂ uˆ + 2αβ ∗ ∂ v∗1 ∂ v1

∂ λˆ ∂ uˆ ∂ uˆ + 6α λˆ 2 ∗ + 12α λˆ ∗ uˆ = 0. ∗ ∂ v1 ∂ v1 ∂ v1

(10.5)

On the other hand we must have also γ 2

1 ∇λˆ · ∇λˆ dx + 2 Ω

Z

Z

6α λˆ 2 uˆ2 dx −

Ω

Z

αβ λˆ 2 dx = 0,

Ω

so that taking the variation in v∗1 for this last equation and considering that −γ∇2 λˆ + 6α uˆ2 λˆ − 2αβ λˆ = 0, we get 12α λˆ 2 uˆ

∂ uˆ = 0. ∂ v∗1

Hence if locally λˆ 2 uˆ ̸= 0, then locally ∂ uˆ = 0. ∂ v∗1 On the other hand if λˆ 2 uˆ = 0, then from (10.5) we have 1 ∂ uˆ = . ∂ v∗1 6α uˆ2 − 2αβ − 6α λˆ 2 + K Recalling that ∂ 2 G∗ (v∗1 ) ∂ uˆ = ∗, ∗ 2 ∂ (v1 ) ∂ v1 we have got ∂ 2 G∗ (v∗1 ) = ∂ (v∗1 )2

(

0, 1 , 6α uˆ2 −2αβ −6α λˆ 2 +K

if λˆ 2 uˆ ̸= 0, if λˆ 2 uˆ = 0.

(10.6)

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Observe also that ∂ 2 J ∗ (v∗1 ) ∂ 2 F ∗ (v∗1 ) ∂ 2 G∗ (v∗1 ) 1 ∂ uˆ = − = − ∗, ∗ ∗ ∗ 2 2 2 2 ∂ (v1 ) ∂ (v1 ) ∂ (v1 ) γ∇ + K ∂ v1 so that, for λˆ 2 uˆ = 0 we obtain 1 ∂ uˆ − γ∇2 + K ∂ v∗1

= = =

1 1 − γ∇2 + K 6α uˆ2 − 2αβ − 6α λˆ 2 + K −γ∇2 − K + 6α uˆ2 − 2αβ − 6α λˆ 2 + K (γ∇2 + K)(6α uˆ2 − 2αβ − 6α λˆ 2 + K) δ 2 J(u) ˆ − 6α λˆ 2

(γ∇2 + K)(6α uˆ2 − 2αβ − 6α λˆ 2 + K) ≥ 0.

(10.7)

Summarizing, ∂ 2 J ∗ (v∗1 ) ∂ (v∗1 )2

=

if λˆ 2 uˆ ̸= 0,

1 , γ∇2 +K

δ 2 J(u)−6α ˆ λˆ 2

(γ∇2 +K)(6α uˆ2 −2αβ −6α λˆ 2 +K)

,

if λˆ 2 uˆ = 0.

(10.8)

Hence, in any case, we have obtained ∂ 2 J ∗ (v∗1 ) ≥ 0, ∀v∗1 ∈ Y ∗ ∂ (v∗1 )2 so that J ∗ is convex in Y ∗ .

10.2

The main duality principle, a convex dual variational formulation

Our main result is summarized by the following theorem. Theorem 10.2.1 Considering the definitions and statements in the last section, suppose also vˆ∗ ∈ Y ∗ is such that δ J ∗ (vˆ∗ ) = 0. Assume also u0 =

∂ F ∗ (vˆ∗1 ) ∈ E + ∩ (A+ )0 . ∂ v∗1

Under such hypotheses, we have δ J(u0 ) = 0,

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and J(u0 ) = = = Proof 10.1

inf {J(u)}

u∈E +

inf J ∗ (v∗1 )

v∗1 ∈Y ∗ J1∗ (vˆ∗1 ).

(10.9)

From the hypothesis ∂ J ∗ (vˆ∗1 ) = 0. ∂ v∗1

so that ∂ J ∗ (vˆ∗1 ) ∂ v∗1

∂ F ∗ (vˆ∗1 ) ∂ G∗1 (vˆ∗1 ) − = 0. ∂ v∗1 ∂ v∗1

=

(10.10)

Since from the previous section we have got that J ∗ is convex on Y ∗ , we may infer that J ∗ (vˆ∗1 ) = ∗inf ∗ J ∗ (v∗1 ). v1 ∈Y

Also, from these last results, u0 −

∂ G∗ (vˆ∗1 ) = 0, ∂ v∗1

so that, since the restriction is not active in a neighborhood of u0 , from the Legendre transform properties, we obtain vˆ∗1 =

∂ G(u0 ) , ∂u

vˆ∗1 =

∂ F(u0 ) , ∂u

and

and thus 0 = vˆ∗1 − vˆ∗1 = −

∂ F(u0 ) ∂ G(u0 ) + = δ J(u0 ). ∂u ∂u

Summarizing δ J(u0 ) = 0. Also from the Legendre transform properties we have F ∗ (vˆ∗1 ) = ⟨u0 , vˆ∗1 ⟩L2 − F(u0 ), and G∗ (vˆ∗1 ) = ⟨u0 , vˆ∗1 ⟩L2 − G(u0 ), so that J ∗ (vˆ∗1 ) = F ∗ (vˆ∗1 ) − G∗ (vˆ∗1 ) = −F(u0 ) + G(u0 ) = J(u0 ).

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Finally, from similar results in [13], we may infer that E + is convex so that from this and δ J(u0 ) = 0, we get J(u0 ) = min J(u). u∈E +

Joining the pieces, we have got J(u0 ) = = = The proof is complete.

inf {J(u)}

u∈E +

inf J ∗ (v∗1 )

v∗1 ∈Y ∗ J ∗ (vˆ∗1 ).

(10.11)

Chapter 11

Duality Principles and Numerical Procedures for a Large Class of Non-Convex Models in the Calculus of Variations

11.1

Introduction

In this section, we establish a dual formulation for a large class of models in nonconvex optimization. The main duality principle is applied to double well models similar as those found in the phase transition theory. Such results are based on the works of J.J. Telega and W.R. Bielski [10, 11, 74, 75], and on a D.C. optimization approach developed in Toland [81]. About the other references, details on the Sobolev spaces involved are found in [1, 26]. Related results on convex analysis and the duality theory are addressed in [12, 13, 14, 22, 23, 25, 62]. Moreover, related results for phase transition and similar models may be found in [29, 33, 36, 37, 38, 44, 45, 47, 56, 57, 58, 59, 63, 64 ,72, 84]. Concerning results in shape optimization may be found in [57, 68] and basic approaches on analysis and functional analysis are developed in [26, 33, 48, 49, 65, 66].

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Finally, in this text we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated. In order to clarify the notation, here we introduce the definition of topological dual space. Definition 11.1.1 (Topological dual spaces) Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined on U. We suppose such a dual space of U, may be represented by another Banach space U ∗ , through a bilinear form ⟨·, ·⟩U : U ×U ∗ → R (here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given f : U → R linear and continuous, we assume the existence of a unique u∗ ∈ U ∗ such that f (u) = ⟨u, u∗ ⟩U , ∀u ∈ U.

(11.1)

The norm of f , denoted by ∥ f ∥U ∗ , is defined as ∥ f ∥U ∗ = sup{|⟨u, u∗ ⟩U | : ∥u∥U ≤ 1} ≡ ∥u∗ ∥U ∗ .

(11.2)

u∈U

At this point we start to describe the primal and dual variational formulations.

11.2

A general duality principle non-convex optimization

In this section, we present a duality principle applicable to a model in phase transition. This case corresponds to the vectorial one in the calculus of variations. Let Ω ⊂ Rn be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω. Consider a functional J : V → R where J(u) = F(∇u1 , · · · , ∇uN ) + G(u1 , · · · , uN ) − ⟨ui , fi ⟩L2 , and where V = {u = (u1 , · · · , uN ) ∈ W 1,p (Ω; RN ) : u = u0 on ∂ Ω}, f ∈ L2 (Ω; RN ), and 1 < p < +∞. We assume there exists α ∈ R such that α = inf J(u). u∈V

Moreover, suppose F and G are Fr´echet differentiable but not necessarily convex. A global optimum point may not be attained for J so that the problem of finding a global minimum for J may not be a solution. Anyway, one question remains, how the minimizing sequences behave close the infimum of J. We intend to use duality theory to approximately solve such a global optimization problem.

Duality Principles and Numerical Procedures for Non-Convex Models

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221

Denoting V0 = W01,p (Ω; RN ), Y1 = Y1∗ = L2 (Ω; RN×n ), Y2 = Y2∗ = L2 (Ω; RN×n ), Y3 = Y3∗ = L2 (Ω; RN ), at this point we define, F1 : V ×V0 → R, G1 : V → R, G2 : V → R, G3 : V0 → R and G4 : V → R, by F1 (∇u, ∇φ ) =

F(∇u1 + ∇φ1 , · · · , ∇uN + ∇φN ) + +

K2 2

Z

K 2

Z

∇u j · ∇u j dx

Ω

∇φ j · ∇φ j dx

(11.3)

Ω

and G1 (u1 , · · · , un ) = G(u1 , · · · , uN ) +

K1 2

Z Ω

u j u j dx − ⟨ui , fi ⟩L2 ,

K1 ∇u j · ∇u j dx, 2 Ω Z K2 G3 (∇φ1 , · · · , ∇φN ) = ∇φ j · ∇φ j dx, 2 Ω Z

G2 (∇u1 , · · · , ∇uN ) =

and G4 (u1 , · · · , uN ) =

K1 2

Z

u j u j dx. Ω

Define now J1 : V ×V0 → R, J1 (u, φ ) = F(∇u + ∇φ ) + G(u) − ⟨ui , fi ⟩L2 . Observe that J1 (u, φ ) = F1 (∇u, ∇φ ) + G1 (u) − G2 (∇u) − G3 (∇φ ) − G4 (u) ≤ F1 (∇u, ∇φ ) + G1 (u) − ⟨∇u, z∗1 ⟩L2 − ⟨∇φ , z∗2 ⟩L2 − ⟨u, z∗3 ⟩L2 + sup {⟨v1 , z∗1 ⟩L2 − G2 (v1 )} v1 ∈Y1

+ sup {⟨v2 , z∗2 ⟩L2 − G3 (v2 )} v2 ∈Y2

+ sup{⟨u, z∗3 ⟩L2 − G4 (u)} u∈V

= F1 (∇u, ∇φ ) + G1 (u) − ⟨∇u, z∗1 ⟩L2 − ⟨∇φ , z∗2 ⟩L2 − ⟨u, z∗3 ⟩L2 +G∗2 (z∗1 ) + G∗3 (z∗2 ) + G∗4 (z∗3 ) = J1∗ (u, φ , z∗ ), (11.4) ∀u ∈ V, φ ∈ V0 , z∗ = (z∗1 , z∗2 , z∗3 ) ∈ Y ∗ = Y1∗ ×Y2∗ ×Y3∗ . Here we assume K, K1 , K2 are large enough so that F1 and G1 are convex. Hence, from the general results in [81], we may infer that inf

(u,φ )∈V ×V0

J(u, φ ) =

inf

(u,φ ,z∗ )∈V ×V0 ×Y ∗

J1∗ (u, φ , z∗ ).

(11.5)

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On the other hand inf J(u) ≥

u∈V

inf

(u,φ )∈V ×V0

J1 (u, φ ).

From these last two results we may obtain inf J(u) ≥

u∈V

inf

(u,φ ,z∗ )∈V ×V0 ×Y ∗

J1∗ (u, φ , z∗ ).

Moreover, from standards results on convex analysis, we may have inf J1∗ (u, φ , z∗ ) =

inf {F1 (∇u, ∇φ ) + G1 (u)

u∈V

u∈V

−⟨∇u, z∗1 ⟩L2 − ⟨∇φ , z∗2 ⟩L2 − ⟨u, z∗3 ⟩L2 +G∗2 (z∗1 ) + G∗3 (z∗2 ) + G∗4 (z∗3 )} = sup {−F1∗ (v∗1 + z∗1 , ∇φ ) − G∗1 (v∗2 + z∗3 ) − ⟨∇φ , z∗2 ⟩L2 (v∗1 ,v∗2 )∈C∗ +G∗2 (z∗1 ) + G∗3 (z∗2 ) + G∗4 (z∗3 )},

(11.6)

where C∗ = {v∗ = (v∗1 , v∗2 ) ∈ Y1∗ ×Y3∗ : − div(v∗1 )i + (v∗2 )i = 0, ∀i ∈ {1, · · · , N}}, F1∗ (v∗1 + z∗1 , ∇φ ) = sup {⟨v1 , z∗1 + v∗1 ⟩L2 − F1 (v1 , ∇φ )}, v1 ∈Y1

and G∗1 (v∗2 + z∗2 ) = sup{⟨u, v∗2 + z∗2 ⟩L2 − G1 (u)}. u∈V

Thus, defining J2∗ (φ , z∗ , v∗ ) = F1∗ (v∗1 +z∗1 , ∇φ )−G∗1 (v∗2 +z∗3 )−⟨∇φ , z∗2 ⟩L2 +G∗2 (z∗1 )+G∗3 (z∗2 )+G∗4 (z∗3 ), we have got inf J(u) ≥

u∈V

= =

inf

(u,φ )∈V ×V0

J1 (u, φ )

inf J1∗ (u, φ , z∗ ) ∗ ∗ ∗ inf inf sup J (φ , z , v ) . 2 ∗ ∗

(u,φ ,z∗ )∈V ×V0 ×Y ∗

z ∈Y

φ ∈V0

(11.7)

v∗ ∈C∗

Finally, observe that inf J(u) ∗ ∗ ∗ ≥ ∗inf ∗ inf sup J2 (φ , z , v ) z ∈Y φ ∈V0 v∗ ∈C∗ ∗ ∗ ∗ ≥ sup inf J2 (φ , z , v ) . u∈V

v∗ ∈C∗

(z∗ ,φ )∈Y ∗ ×V0

(11.8)

This last variational formulation corresponds to a concave relaxed formulation in v∗ concerning the original primal formulation.

Duality Principles and Numerical Procedures for Non-Convex Models

11.3

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223

Another duality principle for a simpler related model in phase transition with a respective numerical example

In this section, we present another duality principle for a related model in phase transition. Let Ω = [0, 1] ⊂ R and consider a functional J : V → R where 1 J(u) = 2

1 ((u ) − 1) dx + 2 Ω

Z

′ 2

2

Z Ω

u2 dx − ⟨u, f ⟩L2 ,

and where V = {u ∈ W 1,4 (Ω) : u(0) = 0 and u(1) = 1/2} and f ∈ L2 (Ω). A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution. Anyway, one question remains, how the minimizing sequences behave close the infimum of J. We intend to use duality theory to approximately solve such a global optimization problem. Denoting V0 = W01,4 (Ω), at this point we define, F : V → R and F1 : V ×V0 → R by F(u) =

1 2

Z

((u′ )2 − 1)2 dx,

Ω

and F1 (u, φ ) =

1 2

Z

((u′ + φ ′ )2 − 1)2 dx.

Ω

Observe that F(u) ≥ inf F1 (u, φ ), ∀u ∈ V. φ ∈V0

In order to restrict the action of φ only on the region where the primal functional is nonconvex, we redefine a not relabeled V0 = {φ ∈ W01,4 (Ω) : (φ ′ )2 ≤ 1, in (Ω)} and we also define F2 : V ×V0 → R, F3 : V ×V0 → R and G : V ×V0 → R by F2 (u, φ ) =

1 2

Z Ω

((u′ + φ ′ )2 − 1)2 dx +

1 2

Z Ω

u2 dx − ⟨u, f ⟩L2 ,

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F3 (u, φ ) =

F2 (u, φ ) + K1 + 2

Z

K 2

Z

(u′ )2 dx

Ω

(φ ′ )2 dx

(11.9)

Ω

and G(u, φ ) =

K 2 +

Z

(u′ )2 dx

Ω

K1 2

Z

(φ ′ )2 dx

(11.10)

Ω

Observe that if K > 0, K1 > 0 is large enough, both F3 and G are convex. Denoting Y = Y ∗ = L2 (Ω) we also define the polar functional G∗ : Y ∗ ×Y ∗ → R by G∗ (v∗ , v∗0 ) =

sup (u,φ )∈V ×V0

{⟨u, v∗ ⟩L2 + ⟨φ , v∗0 ⟩L2 − G(u, φ )}.

Observe that inf J(u) ≥

u∈U

inf

((u,φ ),(v∗ ,v∗0 ))∈V ×V0 ×[Y ∗ ]2

{G∗ (v∗ , v∗0 ) − ⟨u, v∗ ⟩L2 − ⟨φ , v∗0 ⟩L2 + F3 (u, φ )}.

With such results in mind, we define a relaxed primal dual variational formulation for the primal problem, represented by J1∗ : V ×V0 × [Y ∗ ]2 → R, where J1∗ (u, φ , v∗ , v∗0 ) = G∗ (v∗ , v∗0 ) − ⟨u, v∗ ⟩L2 − ⟨φ , v∗0 ⟩L2 + F3 (u, φ ). Having defined such a functional, we may obtain numerical results by solving a sequence of convex auxiliary sub-problems, through the following algorithm. (Here we highlight at first to have neglected the restriction (φ ′ )2 ≤ 1 in (Ω) to obtain the concerning critical points.) 1. Set K = 0.1, K1 = 120 and 0 < ε ≪ 1. 2. Choose (u1 , φ1 ) ∈ V ×V0 , such that ∥u1 ∥1,∞ < 1 and ∥φ1 ∥1,∞ < 1. 3. Set n = 1. 4. Calculate (v∗n , (v∗0 )n ) solution of the system of equations: ∂ J1∗ (un , φn , v∗n , (v∗0 )n ) =0 ∂ v∗ and

that is

∂ J1∗ (un , φn , v∗n , (v∗0 )n ) = 0, ∂ v∗0 ∂ G∗ (v∗n , (v∗0 )n ) − un = 0 ∂ v∗

Duality Principles and Numerical Procedures for Non-Convex Models

and

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225

∂ G∗ (v∗n , (v∗0 )n ) − φn = 0 ∂ v∗0

so that v∗n =

∂ G(un , φn ) ∂u

and (v∗0 )∗n =

∂ G(un , φn ) ∂φ

5. Calculate (un+1 , φn+1 ) by solving the system of equations: ∂ J1∗ (un+1 , φn+1 , v∗n , (v∗0 )n ) =0 ∂u and

∂ J1∗ (un+1 , φn+1 , v∗n , (v∗0 )n ) =0 ∂φ

that is −v∗n +

∂ F3 (un+1 , φn+1 ) =0 ∂u

and −(v∗0 )n +

∂ F3 (un+1 , φn+1 ) =0 ∂φ

6. If max{∥un − un+1 ∥∞ , ∥φn+1 − φn ∥∞ } ≤ ε, then stop, else set n := n + 1 and go to item 4. For the case in which f (x) = 0, we have obtained numerical results for K = 1500 and K1 = K/20. For such a concerning solution u0 obtained, please see Figure 11.1. For the case in which f (x) = sin(πx)/2, we have obtained numerical results for K = 100 and K1 = K/20. For such a concerning solution u0 obtained, please see Figure 11.2. Remark 11.3.1 Observe that the solutions obtained are approximate critical points. They are not, in a classical sense, the global solutions for the related optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

11.4

A convex dual variational formulation for a third similar model

In this section, we present another duality principle for a third related model in phase transition. Let Ω = [0, 1] ⊂ R and consider a functional J : V → R where J(u) =

1 2

Z Ω

min{(u′ − 1)2 , (u′ + 1)2 } dx +

1 2

Z Ω

u2 dx − ⟨u, f ⟩L2 ,

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The Method of Lines and Duality Principles for Non-Convex Models 0.5

0.4

0.3

0.2

0.1

0

-0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11.1: Solution u0 (x) for the case f (x) = 0.

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11.2: Solution u0 (x) for the case f (x) = sin(πx)/2.

and where V = {u ∈ W 1,2 (Ω) : u(0) = 0 and u(1) = 1/2} and f ∈ L2 (Ω). A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution. Anyway, one question remains, how the minimizing sequences behave close to the infimum of J. We intend to use the duality theory to solve such a global optimization problem in an appropriate sense to be specified.

Duality Principles and Numerical Procedures for Non-Convex Models

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227

At this point we define, F : V → R and G : V → R by 1 min{(u′ − 1)2 , (u′ + 1)2 } dx 2 Ω Z Z 1 (u′ )2 dx − |u′ | dx + 1/2 = 2 Ω Ω ≡ F1 (u′ ), Z

F(u) =

and G(u) = G∗

1 2

Z Ω

(11.11)

u2 dx − ⟨u, f ⟩L2 .

Denoting Y = Y ∗ = L2 (Ω) we also define the polar functional F1∗ : Y ∗ → R and : Y ∗ → R by F1∗ (v∗ ) =

sup{⟨v, v∗ ⟩L2 − F1 (v)} v∈Y

1 2

=

Z

(v∗ )2 dx +

Ω

Z

|v∗ | dx,

(11.12)

Ω

and G∗ ((v∗ )′ ) =

sup{−⟨u′ , v∗ ⟩L2 − G(u)} u∈V

=

1 2

Z Ω

1 ((v∗ )′ + f )2 dx − v∗ (1). 2

(11.13)

Observe this is the scalar case of the calculus of variations, so that from the standard results on convex analysis, we have inf J(u) = max {−F1∗ (v∗ ) − G∗ (−(v∗ )′ )}. ∗ ∗

u∈V

v ∈Y

Indeed, from the direct method of the calculus of variations, the maximum for the dual formulation is attained at some vˆ∗ ∈ Y ∗ . Moreover, the corresponding solution u0 ∈ V is obtained from the equation u0 =

∂ G((vˆ∗ )′ ) = (vˆ∗ )′ + f . ∂ (v∗ )′

Finally, the Euler-Lagrange equations for the dual problem stands for ∗ ′′ in Ω, (v ) + f ′ − v∗ − sign(v∗ ) = 0, (v∗ )′ (0) + f (0) = 0, (v∗ )′ (1) + f (1) = 1/2, where sign(v∗ (x)) = 1 if v∗ (x) > 0, sign(v∗ (x)) = −1, if v∗ (x) < 0 and −1 ≤ sign(v∗ (x)) ≤ 1, if v∗ (x) = 0.

(11.14)

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0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11.3: Solution u0 (x) for the case f (x) = 0.

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11.4: Solution u0 (x) for the case f (x) = sin(πx)/2.

We have computed the solutions v∗ and corresponding solutions u0 ∈ V for the cases in which f (x) = 0 and f (x) = sin(πx)/2. For the solution u0 (x) for the case in which f (x) = 0, please see Figure 11.3. For the solution u0 (x) for the case in which f (x) = sin(πx)/2, please see Figure 11.4. Remark 11.4.1 Observe that such solutions u0 obtained are not the global solutions for the related primal optimization problems. Indeed, such solutions reflect the average behavior of weak cluster points for concerning minimizing sequences.

Duality Principles and Numerical Procedures for Non-Convex Models

11.4.1

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229

The algorithm through which we have obtained the numerical results

In this subsection we present the software in MATLAB® through which we have obtained the last numerical results. This algorithm is for solving the concerning Euler-Lagrange equations for the dual problem, that is, for solving the equation ∗ ′′ (v ) + f ′ − v∗ − sign(v∗ ) = 0, in Ω, (11.15) (v∗ )′ (0) = 0, (v∗ )′ (1) = 1/2. Here the concerning software in MATLAB. We emphasize to have used the smooth approximation q |v∗ | ≈ (v∗ )2 + e1 , where a small value for e1 is specified in the next lines. ************************************* 1. clear all 2. m8 = 800; (number of nodes) 3. d = 1/m8 ; 4. e1 = 0.00001; 5. f or i = 1 : m8 yo(i, 1) = 0.01; y1 (i, 1) = sin(π ∗ i/m8 )/2; end; 6. f or i = 1 : m8 − 1 dy1 (i, 1) = (y1 (i + 1, 1) − y1 (i, 1))/d; end; 7. f or k = 1 : 3000 (we have fixed the number of iterations) i = 1; p h3 = 1/ vo(i, 1)2 + e1 ; m12 = 1 + d 2 ∗ h3 + d 2 ; m50 (i) = 1/m12 ; z(i) = m50 (i) ∗ (dy1 (i, 1) ∗ d 2 );

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8. f or i = 2 : m8 − 1 p h3 = 1/ vo(i, 1)2 + e1 ; m12 = 2 + h3 ∗ d 2 + d 2 − m50(i − 1); m50(i) = 1/m12 ; z(i) = m50 (i) ∗ (z(i − 1) + dy1 (i, 1) ∗ d 2 ); end; 9. v(m8 , 1) = (d/2 + z(m8 − 1))/(1 − m50 (m8 − 1)); 10. f or i = 1 : m8 − 1 v(m8 − i, 1) = m50 (m8 − i) ∗ v(m8 − i + 1) + z(m8 − i); end; 11. v(m8 /2, 1) 12. vo = v; end; 13. f or i = 1 : m8 − 1 u(i, 1) = (v(i + 1, 1) − v(i, 1))/d + y1 (i, 1); end; 14. f or i = 1 : m8 − 1 x(i) = i ∗ d; end; plot(x, u(:, 1)) ********************************

11.5

An improvement of the convexity conditions for a non-convex related model through an approximate primal formulation

In this section, we develop an approximate primal dual formulation suitable for a large class of variational models. Here, the applications are for the Kirchhoff-Love plate model, which may be found in Ciarlet [31]. At this point, we start to describe the primal variational formulation. Let Ω ⊂ R2 be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of Ω, which is assumed to be regular (Lipschitzian), is denoted by ∂ Ω. The vectorial basis related to the cartesian system {x1 , x2 , x3 } is denoted by (aα , a3 ), where α = 1, 2 (in general Greek indices stand for

Duality Principles and Numerical Procedures for Non-Convex Models

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231

1 or 2), and where a3 is the vector normal to Ω, whereas a1 and a2 are orthogonal vectors parallel to Ω. Also, n is the outward normal to the plate surface. The displacements will be denoted by uˆ = {uˆα , uˆ3 } = uˆα aα + uˆ3 a3 . The Kirchhoff-Love relations are uˆα (x1 , x2 , x3 ) = uα (x1 , x2 ) − x3 w(x1 , x2 ),α and uˆ3 (x1 , x2 , x3 ) = w(x1 , x2 ).

(11.16)

Here −h/2 ≤ x3 ≤ h/2 so that we have u = (uα , w) ∈ U where U = u = (uα , w) ∈ W 1,2 (Ω; R2 ) ×W 2,2 (Ω), ∂w = 0 on ∂ Ω} uα = w = ∂n = W01,2 (Ω; R2 ) ×W02,2 (Ω). It is worth emphasizing that the boundary conditions specified here refer to a clamped plate. We also define the operator Λ : U → Y ×Y , where Y = Y ∗ = L2 (Ω; R2×2 ), by Λ(u) = {γ(u), κ(u)}, uα,β + uβ ,α w,α w,β + , 2 2 καβ (u) = −w,αβ .

γαβ (u) =

The constitutive relations are given by Nαβ (u) = Hαβ λ µ γλ µ (u),

(11.17)

Mαβ (u) = hαβ λ µ κλ µ (u), (11.18) o n 2 h where: {Hαβ λ µ } and hαβ λ µ = 12 Hαβ λ µ , are symmetric positive definite fourth order tensors. From now on, we denote {H αβ λ µ } = {Hαβ λ µ }−1 and {hαβ λ µ } = {hαβ λ µ }−1 . Furthermore {Nαβ } denote the membrane force tensor and {Mαβ } the moment one. The plate stored energy, represented by (G ◦ Λ) : U → R is expressed by (G ◦ Λ)(u) =

1 2

Z

Nαβ (u)γαβ (u) dx + Ω

1 2

Z

Mαβ (u)καβ (u) dx

(11.19)

Ω

and the external work, represented by F : U → R, is given by F(u) = ⟨w, P⟩L2 + ⟨uα , Pα ⟩L2 ,

(11.20)

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The Method of Lines and Duality Principles for Non-Convex Models

where P, P1 , P2 ∈ L2 (Ω) are external loads in the directions a3 , a1 and a2 respectively. The potential energy, denoted by J : U → R is expressed by: J(u) = (G ◦ Λ)(u) − F(u) Define now J3 : U˜ → R by J3 (u) = J(u) + J5 (w). where aK b w dx + 10 Ω ln(a) K 3/2

Z

J5 (w) = 10

a−K(b w−1/100) dx. Ω ln(a) K 3/2

Z

In such a case for a = 2.71, K = 185, b = P/|P| in Ω and U˜ = {u ∈ U : ∥w∥∞ ≤ 0.01 and P w ≥ 0 a.e. in Ω}, we get ∂ J3 (u) ∂w

= ≈

∂ J(u) ∂ J5 (u) + ∂w ∂w ∂ J(u) + O(±3.0), ∂w

(11.21)

and ∂ 2 J3 (u) ∂ w2

= ≈

∂ 2 J(u) ∂ 2 J5 (u) + ∂ w2 ∂ w2 2 ∂ J(u) + O(850). ∂ w2

(11.22)

This new functional J3 has a relevant improvement in the convexity conditions concerning the previous functional J. 2 , Indeed, we have obtained a gain in positiveness for the second variation ∂ ∂ J(u) w2 which has increased of order O(700 − 1000). Moreover, the difference between the approximate and exact equation ∂ J(u) =0 ∂w is of order O(±3.0) which corresponds to a small perturbation in the original equation for a load of P = 1500 N/m2 , for example. Summarizing, the exact equation may be approximately solved in an appropriate sense.

11.6

An exact convex dual variational formulation for a non-convex primal one

In this section, we develop a convex dual variational formulation suitable to compute a critical point for the corresponding primal one.

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233

Let Ω ⊂ R2 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω. Consider a functional J : V → R where J(u) = F(ux , uy ) − ⟨u, f ⟩L2 , V = W01,2 (Ω) and f ∈ L2 (Ω). Here we denote Y = Y ∗ = L2 (Ω) and Y1 = Y1∗ = L2 (Ω) × L2 (Ω). Defining V1 = {u ∈ V : ∥u∥1,∞ ≤ K1 } for some appropriate K1 > 0, suppose also F is twice Fr´echet differentiable and 2 ∂ F(ux , uy ) ̸= 0, det ∂ v1 ∂ v2 ∀u ∈ V1 . Define now F1 : V → R and F2 : V → R by F1 (ux , uy ) = F(ux , uy ) + and F2 (ux , uy ) =

ε 2

Z Ω

ε 2

Z

u2x dx +

Ω

u2x dx +

ε 2

Z

ε 2

Z

u2y dx,

Ω

u2y dx,

Ω

where here we denote dx = dx1 dx2 . Moreover, we define the respective Legendre transform functionals F1∗ and F2∗ as F1∗ (v∗ ) = ⟨v1 , v∗1 ⟩L2 + ⟨v2 , v∗2 ⟩L2 − F1 (v1 , v2 ), where v1 , v2 ∈ Y are such that v∗1 =

∂ F1 (v1 , v2 ) , ∂ v1

v∗2 =

∂ F1 (v1 , v2 ) , ∂ v2

and F2∗ (v∗ ) = ⟨v1 , v∗1 + f1 ⟩L2 + ⟨v2 , v∗2 ⟩L2 − F2 (v1 , v2 ), where v1 , v2 ∈ Y are such that v∗1 + f1 = v∗2 =

∂ F2 (v1 , v2 ) , ∂ v1

∂ F2 (v1 , v2 ) . ∂ v2

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Here f1 is any function such that ( f1 )x = f , in Ω. Furthermore, we define J ∗ (v∗ ) =

−F1∗ (v∗ ) + F2∗ (v∗ ) Z Z 1 1 = −F1∗ (v∗ ) + (v∗1 + f1 )2 dx + (v∗ )2 dx. 2ε Ω 2ε Ω 2

(11.23)

Observe that through the target conditions v∗1 + f1 = εux , v∗2 = εuy , we may obtain the compatibility condition (v∗1 + f1 )y − (v∗2 )x = 0. Define now A∗ = {v∗ = (v∗1 , v∗2 ) ∈ Br (0, 0) ⊂ Y1∗ : (v∗1 + f1 )y − (v∗2 )x = 0, in Ω}, for some appropriate r > 0 such that J ∗ is convex in Br (0, 0). Consider the problem of minimizing J ∗ subject to v∗ ∈ A∗ . Assuming r > 0 is large enough so that the restriction in r is not active, at this point we define the associated Lagrangian J1∗ (v∗ , ϕ) = J ∗ (v∗ ) + ⟨ϕ, (v∗1 + f )y − (v∗2 )x ⟩L2 , where ϕ is an appropriate Lagrange multiplier. Therefore J1∗ (v∗ )

=

1 1 (v∗ + f1 )2 dx + 2ε Ω 1 2ε +⟨ϕ, (v∗1 + f )y − (v∗2 )x ⟩L2 .

−F1∗ (v∗ ) +

Z

Z

(v∗2 )2 dx

Ω

(11.24)

The optimal point in question will be a solution of the corresponding EulerLagrange equations for J1∗ . From the variation of J1∗ in v∗1 we obtain −

∂ F1∗ (v∗ ) v∗1 + f ∂ ϕ + − = 0. ∂ v∗1 ε ∂y

(11.25)

From the variation of J1∗ in v∗2 we obtain −

∂ F1∗ (v∗ ) v∗2 ∂ ϕ + + = 0. ∂ v∗2 ε ∂x

(11.26)

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235

From the variation of J1∗ in ϕ we have (v∗1 + f )y − (v∗2 )x = 0. From this last equation, we may obtain u ∈ V such that v∗1 + f = εux , and v∗2 = εuy . From this and the previous extremal equations indicated we have −

∂ F1∗ (v∗ ) ∂ϕ + ux − = 0, ∗ ∂ v1 ∂y

−

∂ F1∗ (v∗ ) ∂ϕ + uy + = 0. ∂ v∗2 ∂x

and

so that v∗1 + f = and v∗2 =

∂ F1 (ux − ϕy , uy + ϕx ) , ∂ v1

∂ F1 (ux − ϕy , uy + ϕx ) . ∂ v2

From this and equation (11.25) and (11.26) we have ∗ ∗ ∗ ∗ ∂ F1 (v ) ∂ F1 (v ) −ε −ε ∗ ∂ v1 ∂ v∗2 x y +(v∗1 + f1 )x + (v∗2 )y = −εuxx − εuyy + (v∗1 )x + (v∗2 )y + f = 0.

(11.27)

Replacing the expressions of v∗1 and v∗2 into this last equation, we have ∂ F1 (ux − ϕy , uy + ϕx ) ∂ F1 (ux − ϕy , uy + ϕx ) + + f = 0, −εuxx − εuyy + ∂ v1 ∂ v2 x y so that ∂ F(ux − ϕy , uy + ϕx ) ∂ F(ux − ϕy , uy + ϕx ) + + f = 0, in Ω. ∂ v1 ∂ v2 x y Observe that if ∇2 ϕ = 0 then there exists uˆ such that u and ϕ are also such that ux − ϕy = uˆx

(11.28)

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and uy + ϕx = uˆy . The boundary conditions for ϕ must be such that uˆ ∈ W01,2 . From this and equation (11.28) we obtain δ J(u) ˆ = 0. Summarizing, we may obtain a solution uˆ ∈ W01,2 of equation δ J(u) ˆ = 0 by minimizing J ∗ on A∗ . Finally, observe that clearly J ∗ is convex in an appropriate large ball Br (0, 0) for some appropriate r > 0.

11.7 Another primal dual formulation for a related model Let Ω ⊂ R3 be an open, bounded and connected set with a regular boundary denoted by ∂ Ω. Consider the functional J : V → R where Z

J(u) =

α γ ∇u · ∇u dx + 2 Ω 2 −⟨u, f ⟩L2 ,

Z

(u2 − β )2 dx

Ω

(11.29)

α > 0, β > 0, γ > 0, V = W01,2 (Ω) and f ∈ L2 (Ω). Denoting Y = Y ∗ = L2 (Ω), define now J1∗ : V ×Y ∗ → R by J1∗ (u, v∗0 ) =

Z

γ ∇u · ∇u dx − ⟨u2 , v∗0 ⟩L2 2 Ω Z K1 (−γ∇2 u + 2v∗0 u − f )2 dx + ⟨u, f ⟩L2 + 2 Ω Z Z 1 (v∗0 )2 dx + β v∗0 dx, + 2α Ω Ω −

Define also A+ = {u ∈ V : u f ≥ 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, and V1 = V2 ∩ A+ for some appropriate K3 > 0 to be specified. Moreover define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K} for some appropriate K > 0 to be specified.

(11.30)

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Observe that, denoting ϕ = −γ∇2 u + 2v∗0 u − f we have

∂ 2 J1∗ (u, v∗0 ) 1 = + 4K1 u2 ∂ (v∗0 )2 α ∂ 2 J1∗ (u, v∗0 ) = γ∇2 − 2v∗0 + K1 (−γ∇2 + 2v∗0 )2 ∂ u2

and

∂ 2 J1∗ (u, v∗0 ) = K1 (2ϕ + 2(−γ∇2 u + 2v∗0 u)) − 2u ∂ u∂ v∗0

so that det{δ 2 J1∗ (u, v∗0 )} = =

2 ∗ 2 ∂ J1 (u, v∗0 ) ∂ 2 J1∗ (u, v∗0 ) ∂ 2 J1∗ (u, v∗0 ) − ∂ (v∗0 )2 ∂ u2 ∂ u∂ v∗0 K1 (−γ∇2 + 2v∗0 )2 γ∇2 + 2v∗0 + 4αu2 − α α −4K12 ϕ 2 − 8K1 ϕ(−γ∇2 + 2v∗0 )u + 8K1 ϕu +4K1 (−γ∇2 u + 2v∗0 u)u.

(11.31)

Observe now that a critical point ϕ = 0 and (−γ∇2 u + 2v∗0 u)u = f u ≥ 0 in Ω. Therefore, for an appropriate large K1 > 0, also at a critical point, we have det{δ 2 J1∗ (u, v∗0 )} = 4K1 f u −

(−γ∇2 + 2v∗0 )2 δ 2 J(u) + K1 > 0. α α

(11.32)

Remark 11.7.1 From this last equation we may observe that J1∗ has a large region of convexity about any critical point (u0 , vˆ∗0 ), that is, there exists a large r > 0 such that J1∗ is convex on Br (u0 , vˆ∗0 ). With such results in mind, we may easily prove the following theorem. Theorem 11.7.2 Assume K1 ≫ max{1, K, K3 } and suppose (u0 , vˆ∗0 ) ∈ V1 ×B∗ is such that δ J1∗ (u0 , vˆ∗0 ) = 0. E∗

Under such hypotheses, there exists r > 0 such that J1∗ is convex in = Br (u0 , vˆ∗0 ) ∩ (V1 × B∗ ), δ J(u0 ) = 0,

and −J(u0 ) = J1 (u0 , vˆ∗0 ) =

inf

(u,v∗0 )∈E ∗

J1∗ (u, v∗0 ).

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The Method of Lines and Duality Principles for Non-Convex Models

A third primal dual formulation for a related model

Let Ω ⊂ R3 be an open, bounded and connected set with a regular boundary denoted by ∂ Ω. Consider the functional J : V → R where Z

Z

α γ ∇u · ∇u dx + 2 Ω 2 −⟨u, f ⟩L2 ,

J(u) =

(u2 − β )2 dx

Ω

(11.33)

α > 0, β > 0, γ > 0, V = W01,2 (Ω) and f ∈ L2 (Ω). Denoting Y = Y ∗ = L2 (Ω), define now J1∗ : V ×Y ∗ ×Y ∗ → R by γ 2

J1∗ (u, v∗0 , v∗1 ) =

Z

∇u · ∇u dx +

Ω

−⟨u, v∗1 ⟩L2 +

1 2

1 2

Z

K u2 dx

Ω

(v∗1 )2 dx ∗ Ω (−2v0 + K)

Z

1 (v∗ − α(u2 − β ))2 dx + ⟨u, f ⟩L2 2(α + ε) Ω 0 Z Z 1 ∗ 2 (v ) dx − β v∗0 dx, − 2α Ω 0 Ω Z

+

(11.34)

where ε > 0 is a small real constant. Define also A+ = {u ∈ V : u f ≥ 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, and V1 = V2 ∩ A+ for some appropriate K3 > 0 to be specified. Moreover, define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K4 } and D∗ = {v∗1 ∈ Y ∗ : ∥v∗1 ∥ ≤ K5 }, for some appropriate real constants K4 , K5 > 0 to be specified. Remark 11.8.1 Define now H1 (u, v∗0 ) = −γ∇2 + 2v∗0 + 4αu2 and Eˆv∗0 = {u ∈ V : H1 (u, v∗0 ) ≥ 0}. For a fixed v∗0 ∈ B∗ , we are going to prove that C∗ = Eˆv∗0 ∩V1 is a convex set.

Duality Principles and Numerical Procedures for Non-Convex Models

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Assume, for a finite dimensional problem version, in a finite differences or finite element context, that −γ∇2 − 2v∗0 ≤ 0, so that for K1 > 0 be sufficiently large, we have −γ∇2 + 2v∗0 − K1 u2 ≤ 0. Observe now that H1 (u, v∗0 ) = −γ∇2 + 2v∗0 − K1 u2 + 4αu2 + K1 u2 . Let u1 , u2 ∈ C∗ and λ ∈ [0, 1]. Thus sign (u1 ) = sign (u2 ) in Ω so that λ |u1 | + (1 − λ )|u2 | = |λ u1 + (1 − λ )u2 | in Ω. Observe now that H1 (u1 , v∗0 ) ≥ 0 and H1 (u2 , v∗0 ) ≥ 0 so that 4αu21 + K1 u21 ≥ γ∇2 − 2v∗0 + K1 u21 ≥ 0, and 4αu22 + K1 u21 ≥ γ∇2 − 2v∗0 + K1 u22 ≥ 0, so that q p 4α + K1 |u1 | ≥ γ∇2 − 2v∗0 + K1 u21 and

q p 4α + K1 |u2 | ≥ γ∇2 − 2v∗0 + K1 u22 . From such results we obtain

p 4α + K1 |λ u1 + (1 − λ )u2 |

= ≥ ≥

p 4α + K1 (λ |u1 | + (1 − λ |u2 |) q q λ γ∇2 − 2v∗0 + K1 u21 + (1 − λ ) γ∇2 − 2v∗0 + K1 u22 q γ∇2 − 2v∗0 + K1 (λ u1 + (1 − λ )u2 )2 . (11.35)

From this we obtain (4α + K1 )(λ u1 + (1 − λ )u2 )2 ≥ γ∇2 − 2v∗0 + K1 (λ u1 + (1 − λ )u2 )2 , so that H1 (λ u1 + (1 − λ )u2 , v∗0 ) ≥ 0. C∗

Hence Eˆv∗0 is convex. Since V1 is also clearly convex, we have obtained that = Eˆv∗0 ∩V1 is convex. Such a result we will be used many times in the next sections.

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Observe that, defining ϕ = v∗0 − α(u2 − β ) we may obtain ∂ 2 J1∗ (u, v∗0 , v∗1 ) α α = −γ∇2 + K + 4u2 − 2ϕ 2 ∂u α +ε α +ε ∂ 2 J1∗ (u, v∗0 , v∗1 ) 1 = ∂ (v∗1 )2 −2v∗0 + K and

∂ 2 J1∗ (u, v∗0 , v∗1 ) = −1 ∂ u∂ v∗1

so that det =

∂ 2 J1∗ (u, v∗0 , v∗1 ) ∂ u∂ v∗1

2 ∗ 2 ∂ J1 (u, v∗1 , v∗0 ) ∂ 2 J1∗ (u, v∗1 , v∗0 ) ∂ 2 J1∗ (u, v∗1 , v∗0 ) − ∂ (v∗1 )2 ∂ u2 ∂ u∂ v∗1 2

α α −γ∇2 + 2v∗0 + 4 α+ε u2 − 2 α+ε ϕ ∗ −2v0 + K ∗ ≡ H(u, v0 ).

=

(11.36)

However, at a critical point, we have ϕ = 0 so that, we define Cv∗∗ = {u ∈ V : ϕ ≤ 0}. 0

From such results, assuming K ≫ max{K3 , K4 , K5 }, define now Ev∗0 = {u ∈ V : H(u, v∗0 ) > 0}. Observe that similarly as it was develop in remark 11.8.1, we may prove that Ev∗0 is a convex set. With such results in mind, we may easily prove the following theorem. Theorem 11.8.2 Suppose (u0 , vˆ∗1 , vˆ∗0 ) ∈ E ∗ = (V1 ∩ Cvˆ∗0 ∩ Evˆ∗0 ) × D∗ × B∗ and −γ∇2 − 2vˆ∗0 ≤ 0 is such that δ J1∗ (u0 , vˆ∗0 , vˆ∗1 ) = 0. Under such hypotheses, we have that δ J(u0 ) = 0

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241

and J(u0 ) = =

inf J(u)

u∈V1 J1∗ (u0 , vˆ∗1 , vˆ∗0 )

( =

inf

(u,v∗1 )∈V1 ×D∗

) sup

v∗0 ∈B∗

J1∗ (u, v∗1 , v∗0 )

( =

sup

inf

(u,v∗1 )∈V1 ×D∗

v∗0 ∈B∗

Proof 11.1

) J1∗ (u, v∗1 , v∗0 ) .

(11.37)

The proof that δ J(u0 ) = 0

and J(u0 ) = J1∗ (u0 , vˆ∗1 , vˆ∗0 ) may be easily made similarly as in the previous sections. Moreover, from the hypotheses, we have J1∗ (u0 , vˆ∗1 , vˆ∗0 ) =

inf

(u,v∗1 )∈V1 ×D∗

J1∗ (u, v∗1 , vˆ∗0 )

and J1∗ (u0 , vˆ∗1 , vˆ∗0 ) = sup J1∗ (u0 , vˆ∗1 , v∗0 ). v∗0 ∈B∗

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that J(u0 ) = =

J1∗ (u0 , vˆ∗1 , vˆ∗0 ) ( inf

(u,v∗1 )∈V1 ×D∗

) sup

v∗0 ∈B∗

( =

sup

v∗0 ∈B∗

J1∗ (u, v∗1 , v∗0 ) )

inf

(u,v∗1 )∈V1 ×D

J ∗ (u, v∗1 , v∗0 ) ∗ 1

.

(11.38)

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Moreover, observe that J1∗ (u0 , vˆ∗1 , vˆ∗0 ) =

inf

(u,v∗1 )∈V1 ×D∗

J1∗ (u, v∗1 , vˆ∗0 )

K ∇u · ∇u dx + u2 dx ≤ 2 Ω Ω Z K +⟨u2 , vˆ∗0 ⟩L2 − u2 dx 2 Ω Z Z 1 (vˆ∗0 )2 dx − β vˆ∗0 dx − 2α Ω Ω Z 1 ∗ (vˆ − α(u2 − β ))2 dx − ⟨u, f ⟩L2 + 2(α + ε) Ω 0 Z γ ∇u · ∇u dx + ⟨u2 , v∗0 ⟩ ≤ sup ∗ 2 ∗ Ω v0 ∈Y γ 2

−

Z

Z

1 2α

+

Z

Z

(v∗0 )2 dx − β

Ω

v∗0 dx

Ω

1 2(α + ε)

Z Ω

(v∗0 − α(u2 − β ))2 dx − ⟨u, f ⟩L2

Z

α γ ∇u · ∇u dx + = 2 Ω 2 −⟨u, f ⟩L2 , ∀u ∈ V1 .

Z

(u2 − β )2 dx

Ω

(11.39)

Summarizing, we have got J(u0 ) = J1∗ (u0 , vˆ∗1 , vˆ∗0 ) ≤ inf J(u). u∈V1

From such results, we may infer that J(u0 ) = =

inf J(u)

u∈V1 J1∗ (u0 , vˆ∗1 , vˆ∗0 )

( =

inf

(u,v∗1 )∈V1 ×D∗

) sup

v∗0 ∈B∗

J1∗ (u, v∗1 , v∗0 )

( =

sup

v∗0 ∈B∗

) inf

(u,v∗1 )∈V1 ×D

J ∗ (u, v∗1 , v∗0 ) ∗ 1

.

(11.40)

The proof is complete.

11.9

A fourth primal dual formulation for a related model

Let Ω ⊂ R3 be an open, bounded and connected set with a regular boundary denoted by ∂ Ω.

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243

Consider the functional J : V → R where Z

α γ ∇u · ∇u dx + 2 Ω 2 −⟨u, f ⟩L2 ,

J(u) =

Z

(u2 − β )2 dx

Ω

(11.41)

α > 0, β > 0, γ > 0, V = W01,2 (Ω) and f ∈ L2 (Ω). Denoting Y = Y ∗ = L2 (Ω), define now J1∗ : V ×Y ∗ → R by J1∗ (u, v∗0 )

=

γ 2

Z Ω

∇u · ∇u dx − ⟨u2 , v∗0 ⟩L2

1 (v∗ − α(u2 − β ))2 dx − ⟨u, f ⟩L2 2(α + ε) Ω 0 Z Z 1 ∗ 2 (v ) dx − β v∗0 dx, − 2α Ω 0 Ω Z

+

(11.42)

where ε > 0 is a small real constant. Define also A+ = {u ∈ V : u f ≥ 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, and V1 = V2 ∩ A+ for some appropriate real constant K3 > 0. Moreover define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K4 } for some appropriate real constant K4 > 0. Observe that, denoting ϕ = v∗0 − α(u2 − β ), we may obtain ∂ 2 J1∗ (u, v∗0 ) ∂ u2

= −γ∇2 + 2v0 α2 ϕ 4u2 − 2 α α +ε α +ε ≡ H(u, v∗0 ), +

and

∂ 2 J1∗ (u, v∗0 ) 1 1 0}.

(11.43)

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Remark 11.9.1 Similarly as it was developed in remark 11.8.1 we may prove that such a Ev∗0 is a convex set. With such results in mind, we may easily prove the following theorem. Theorem 11.9.2 Suppose (u0 , vˆ∗0 ) ∈ E ∗ = (V1 ∩Cvˆ∗0 ∩ Evˆ∗0 ) × B∗ is such that δ J1∗ (u0 , vˆ∗0 ) = 0. Under such hypotheses, we have that δ J(u0 ) = 0 and J(u0 ) = =

inf J(u)

u∈V1 J1∗ (u0 , vˆ∗0 )

( =

inf

u∈V1

) sup

v∗0 ∈B∗

=

sup

v∗0 ∈B∗

Proof 11.2

J1∗ (u, v∗0 )

inf J1∗ (u, v∗0 ) .

u∈V1

(11.44)

The proof that δ J(u0 ) = 0

and J(u0 ) = J1∗ (u0 , vˆ∗0 ) may be easily made similarly as in the previous sections. Moreover, from the hypotheses, we have J1∗ (u0 , vˆ∗0 ) = inf J1∗ (u, vˆ∗0 ) u∈V1

and J1∗ (u0 , vˆ∗0 ) = sup J1∗ (u0 , v∗0 ). v∗0 ∈B∗

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that J(u0 ) = =

J1∗ (u0 , vˆ∗0 ) ( inf

u∈V1

sup

v∗0 ∈B∗

=

sup

v∗0 ∈B∗

) J1∗ (u, v∗0 )

inf J1∗ (u, v∗0 ) .

u∈V1

(11.45)

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Moreover, observe that J1∗ (u0 , vˆ∗0 ) =

inf J1∗ (u, vˆ∗0 )

u∈V1

γ 2

≤

Z Ω

1 − 2α

∇u · ∇u dx + ⟨u2 , vˆ∗0 ⟩L2 Z

(vˆ∗0 )2

dx − β

Z

Ω

vˆ∗0 dx

Ω

1 (vˆ∗ − α(u2 − β ))2 dx − ⟨u, f ⟩L2 2(α + ε) Ω 0 Z γ ∇u · ∇u dx + ⟨u2 , v∗0 ⟩ ≤ sup ∗ v0 ∈Y ∗ 2 Ω Z

+

−

1 2α

Z

(v∗0 )2 dx − β

Z

Ω

v∗0 dx

Ω

1 (v∗ − α(u2 − β ))2 dx − ⟨u, f ⟩L2 + 2(α + ε) Ω 0 Z Z α γ ∇u · ∇u dx + (u2 − β )2 dx = 2 Ω 2 Ω −⟨u, f ⟩L2 , ∀u ∈ V1 . Z

(11.46)

Summarizing, we have got J(u0 ) = J1∗ (u0 , vˆ∗0 ) ≤ inf J(u). u∈V1

From such results, we may infer that J(u0 ) = =

inf J(u)

u∈V1 J1∗ (u0 , vˆ∗0 )

( =

inf

u∈V1

) sup

v∗0 ∈B∗

=

sup

v∗0 ∈B∗

J1∗ (u, v∗0 )

inf J1∗ (u, v∗0 ) .

u∈V1

(11.47)

The proof is complete.

11.10

One more primal dual formulation for a related model

Let Ω ⊂ R3 be an open, bounded and connected set with a regular boundary denoted by ∂ Ω.

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Consider the functional J : V → R where Z Z α γ ∇u · ∇u dx + (u2 − β )2 dx J(u) = 2 Ω 2 Ω −⟨u, f ⟩L2 ,

(11.48)

α > 0, β > 0, γ > 0, V = W01,2 (Ω) and f ∈ L2 (Ω). Denoting Y = Y ∗ = L2 (Ω), define now J1∗ : V ×Y ∗ ×Y ∗ → R by J1∗ (u, v∗1 , v∗0 ) =

γ 2

Z

∇u · ∇u dx +

Ω

K 2

Z Ω

u2 dx − ⟨u, v∗1 ⟩L2

(v∗1 )2 dx − ⟨u, f ⟩L2 ∗ Ω −2v0 + K 2 Z v∗1 v∗1 + f K2 − dx + 2 Ω −γ∇2 + K −2v∗0 + K Z Z 1 − (v∗0 )2 dx − β v∗0 dx, 2α Ω Ω 1 + 2

Z

(11.49)

Define also A+ = {u ∈ V : u f ≥ 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, and V1 = V2 ∩ A+ q

1 . specifically for a constant K3 = 5α Moreover define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K4 }

and D∗ = {v∗1 ∈ Y ∗ : ∥v∗1 ∥∞ ≤ K5 } for some appropriate real constants K4 > 0 and K5 > 0. Observe that ∂ 2 J1∗ (u, v∗1 , v∗0 ) = −γ∇2 + K, ∂ u2 ∂ 2 J1∗ (u, v∗1 , v∗0 ) K 2 (−γ∇2 + 2v∗0 )2 1 + = , ∂ (v∗1 )2 −2v∗0 + K [(−γ∇2 + K)(−2v∗0 + K)]2 ∂ 2 J1∗ (u, v∗1 , v∗0 ) = −1, ∂ u ∂ v∗1 so that 2 ∗ ∂ J1 (u, v∗1 , v∗0 ) det ∂ u ∂ v∗1

2 ∗ 2 ∂ 2 J1∗ (u, v∗1 , v∗0 ) ∂ 2 J1∗ (u, v∗1 , v∗0 ) ∂ J1 (u, v∗1 , v∗0 ) − ∂ (v∗1 )2 ∂ u2 ∂ u ∂ v∗1 2 K (2(−γ∇2 + 2v∗0 ) + 2(−γ∇2 + 2v∗0 )2 ) = O (−γ∇2 + K)(−2v∗0 + K)2 ∗ ≡ H(v0 ). (11.50) =

Duality Principles and Numerical Procedures for Non-Convex Models

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247

With such results in mind, we may easily prove the following theorem. Theorem 11.10.1 Assume K ≫ max{K3 , K4 , K5 , 1} and suppose (u0 , vˆ∗1 , vˆ∗0 ) ∈ V1 × D∗ × B∗ is such that δ J1∗ (u0 , vˆ∗1 , vˆ∗0 ) = 0. Suppose also H(vˆ∗0 ) > 0. Under such hypotheses, we have that δ J(u0 ) = 0 and ( J(u0 ) =

K2 J(u) + 2

inf

u∈V1

Z Ω

(−γ∇2 u + 2vˆ∗0 u − f ) −γ∇2 + K

= J1∗ (u0 , vˆ∗1 , vˆ∗0 ) ( =

inf

sup

v∗0 ∈B∗

Proof 11.3

) inf

(u,v∗1 )∈V1 ×D

v∗0 ∈B∗

dx

J1∗ (u, , v∗1 , v∗0 )

( =

)

) sup

(u,v∗1 )∈V1 ×D∗

2

J ∗ (u, v∗1 , v∗0 ) ∗ 1

.

(11.51)

The proof that δ J(u0 ) = −γ∇2 u0 + 2α(u2 − β )u0 − f = 0, vˆ∗0 = α(u20 − β )

and J(u0 ) = J(u0 ) +

K2 2

Z Ω

(−γ∇2 u0 + 2vˆ∗0 u0 − f ) −γ∇2 + K

2

dx = J1∗ (u0 , vˆ∗1 , vˆ∗0 )

may be easily made similarly as in the previous sections. Moreover, from the hypotheses, we have J1∗ (u0 , vˆ∗1 , vˆ∗0 ) =

inf

(u,v∗1 )∈V1 ×D∗

J1∗ (u, v∗1 , vˆ∗0 )

and J1∗ (u0 , vˆ∗1 , vˆ∗0 ) = sup J1∗ (u0 , vˆ∗1 , v∗0 ). v∗0 ∈B∗

From this, from a standard saddle point theorem and the remaining hypotheses, we may infer that J(u0 ) = =

J1∗ (u0 , vˆ∗1 , vˆ∗0 ) ( inf

(u,v∗1 )∈V1 ×D∗

) sup J1∗ (u, v∗1 , v∗0 )

v∗0 ∈B∗

( =

sup

v∗0 ∈B∗

) inf

(u,v∗1 )∈V1 ×D

J ∗ (u, v∗1 , v∗0 ) ∗ 1

.

(11.52)

248

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The Method of Lines and Duality Principles for Non-Convex Models

Moreover, observe that J1∗ (u0 , vˆ∗1 , vˆ∗0 ) =

inf

(u,v∗1 )∈V1 ×D∗

Z

γ 2

≤

Ω

− +

J1∗ (u, v∗1 , vˆ∗0 )

∇u · ∇u dx + ⟨u2 , vˆ∗0 ⟩L2

1 2α

Z

K2 2

Z

(vˆ∗0 )2 dx − β

Z

vˆ∗0 dx − ⟨u, f ⟩L2 Ω 2 (−γ∇2 u + 2vˆ∗0 u − f ) dx −γ∇2 + K

Ω

Ω

Z γ ≤ sup ∇u · ∇u dx + ⟨u2 , v∗0 ⟩ ∗ 2 ∗ Ω v0 ∈Y 1 2α

Z

(v∗0 )2 dx − β

Z

v∗0 dx − ⟨u, f ⟩L2 2 ) Z (−γ∇2 u + 2vˆ∗0 u − f ) K2 dx + 2 Ω −γ∇2 + K

−

Ω

Ω

Z

Z

Z

Z

γ α ∇u · ∇u dx + (u2 − β )2 dx 2 Ω 2 Ω −⟨u, f ⟩L2 2 Z (−γ∇2 u + 2vˆ∗0 u − f ) K2 dx, ∀u ∈ V1 . (11.53) + 2 Ω −γ∇2 + K

=

From this we have got J1∗ (u0 , vˆ∗1 , vˆ∗0 )

γ 2

α ≤ ∇u · ∇u dx + (u2 − β )2 dx − ⟨u, f ⟩L2 2 Ω Ω 2 Z (−γ∇2 u + 2vˆ∗0 u − f ) K2 dx, ∀u ∈ V1 . (11.54) + 2 Ω −γ∇2 + K

Therefore, from such results we may obtain ( 2 ) Z (−γ∇2 u + 2vˆ∗0 u − f ) K2 J(u0 ) = inf J(u) + dx u∈V1 2 Ω −γ∇2 + K = J1∗ (u0 , vˆ∗1 , vˆ∗0 ) ( =

inf

(u,v∗1 )∈V1 ×D∗

) sup

v∗0 ∈B∗

( =

sup

v∗0 ∈B∗

The proof is complete.

J1∗ (u, , v∗1 , v∗0 ) )

inf

(u,v∗1 )∈V1 ×D

J ∗ (u, v∗1 , v∗0 ) ∗ 1

.

(11.55)

Duality Principles and Numerical Procedures for Non-Convex Models

11.11

■

249

Another primal dual formulation for a related model

In this section, we present another primal dual formulation. Let Ω ⊂ R3 be an open, bounded and connected set with a regular boundary denoted by ∂ Ω. Consider the functional J : V → R where Z

α γ ∇u · ∇u dx + 2 Ω 2 −⟨u, f ⟩L2 ,

J(u) =

Z

(u2 − β )2 dx

Ω

(11.56)

α > 0, β > 0, γ > 0, V = W01,2 (Ω) and f ∈ L2 (Ω). Denoting Y = Y ∗ = L2 (Ω), define now J1∗ : V ×Y ∗ → R by J1∗ (u, v∗0 ) =

γ 2

Z Ω

∇u · ∇u dx + ⟨u2 , v∗0 ⟩L2

α −ε u4 dx − ⟨u, f ⟩L2 2 Ω Z 1 (v∗ + αβ )2 dx, − 2ε Ω 0 Z

+

(11.57)

and J2∗ : V ×Y ∗ → R, by J2∗ (u, v∗0 ) =

γ 2

Z Ω

∇u · ∇u dx + ⟨u2 , v∗0 ⟩L2

K1 (−γ∇2 u + 2v∗0 u − 2(α − ε)u3 − f )2 dx 2 Ω Z α −ε u4 dx − ⟨u, f ⟩L2 + 2 Ω Z 1 − (v∗ + αβ )2 dx, 2ε Ω 0 Z

+

Define also A+ = {u ∈ V : u f ≥ 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, and V1 = V2 ∩ A+ . Moreover define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K4 } for some appropriate constants K3 > 0 and K4 > 0.

(11.58)

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The Method of Lines and Duality Principles for Non-Convex Models

√ Observe that, for K1 = 1/ ε, we have ∂ 2 J2∗ (u, v∗0 ) ∂ u2

= (−γ∇2 + 2v∗0 + 6(α − ε)u2 ) + K1 (−γ∇2 + 2v∗0 + 6(α − ε)u2 )2 +K1 (−γ∇2 u + 2v∗0 u + 2(α − ε)u3 − f )12(α − ε)u dx, (11.59) ∂ 2 J2∗ (u, v∗0 ) ∂ (v∗0 )2

1 ε < 0, ∀u ∈ V1 , v∗0 ∈ B∗ .

= K1 4u2 −

(11.60)

Define now A2 (u, v∗0 ) = (−γ∇2 u + 2v∗0 u + 2(α − ε)u3 − f )12(α − ε)u, C∗ = {(u, v∗0 ) ∈ V × B∗ : ∥A2 (u, v∗0 )∥∞ ≤ ε1 for a small real parameter ε1 > 0. Finally, define

∂ 2 A2 (u, v∗0 ) >0 . E = u∈V : ∂ u2 v∗0

Remark 11.11.1 Similarly as it was developed in remark 11.8.1, we may prove that such a Ev∗0 is a convex set. Thus, Ev∗0 ∩V1 is a convex set, ∀v∗0 ∈ B∗ (for the proof of a similar result please see Theorem 8.7.1 at pages 297, 298 and 299 in [13]).) With such results in mind, we may easily prove the following theorem. Theorem 11.11.2 Assume K1 ≫ 1 ≫ ε1 and suppose (u0 , vˆ∗0 ) ∈ V1 × B∗ is such that δ J2∗ (u0 , vˆ∗0 ) = 0 and u0 ∈ Evˆ∗0 . Under such hypotheses, we have that δ J(u0 ) = 0 and Z K1 2 ∗ 3 2 J(u0 ) = inf J(u) + (−γ∇ u + 2vˆ0 u + 2(α − ε)u − f ) dx u∈V1 2 Ω = J2∗ (u0 , vˆ∗0 ) = sup inf J2∗ (u, v∗0 ) . (11.61) v∗0 ∈B∗

u∈V1

Duality Principles and Numerical Procedures for Non-Convex Models

Proof 11.4

■

251

The proof that δ J(u0 ) = −γ∇2 u0 + 2α(u2 − β )u0 − f = 0,

and J(u0 ) = J(u0 ) +

K1 2

Z

(−γ∇2 u0 + 2vˆ∗0 u0 + 2(α − ε)u30 − f )2 dx = J2∗ (u0 , vˆ∗0 )

Ω

may be easily made similarly as in the previous sections. Moreover, from the hypotheses and from the above lines, since J2∗ is concave in ∗ v0 on V1 × B∗ and u0 ∈ Evˆ∗0 , we have that J2∗ (u0 , vˆ∗0 ) = inf J2∗ (u, vˆ∗0 ) u∈V1

and J2∗ (u0 , vˆ∗0 ) = sup J2∗ (u0 , v∗0 ). v∗0 ∈B∗

From this, from the standard Saddle Point Theorem and the remaining hypotheses, we may infer that J(u0 ) = =

J2∗ (u0 , vˆ∗0 ) ( inf

u∈V1

sup

v∗0 ∈B∗

=

sup

v∗0 ∈B∗

) J2∗ (u, v∗1 , v∗0 )

inf J2∗ (u, v∗0 ) .

u∈V1

(11.62)

Moreover, observe that J2∗ (u0 , vˆ∗0 ) = ≤

inf J2∗ (u, vˆ∗0 )

u∈V1

γ 2

α −ε 2

Z

u4 dx

Ω

Z

1 (v∗ + αβ )2 dx − ⟨u, f ⟩L2 2ε Ω 0 Z K1 2 ∗ 3 2 (−γ∇ u + 2vˆ0 u + 2(α − ε)u − f ) dx + 2 Ω Z K1 (−γ∇2 u + 2vˆ∗0 u + 2(α − ε)u3 − f )2 dx, ∀u ∈ V1 . (11.63) J(u) + 2 Ω

−

=

Ω

∇u · ∇u dx + ⟨u2 , vˆ∗0 ⟩L2 +

1 (vˆ∗ + αβ )2 dx − ⟨u, f ⟩L2 2ε Ω 0 Z K1 (−γ∇2 u + 2vˆ∗0 u + 2(α − ε)u3 − f )2 dx + 2 Ω Z Z α −ε γ ∇u · ∇u dx + ⟨u2 , v∗0 ⟩L2 + u4 dx sup 2 Ω v∗0 ∈Y ∗ 2 Ω

−

≤

Z

Z

252

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The Method of Lines and Duality Principles for Non-Convex Models

From this we have got J2∗ (u0 , vˆ∗0 )

≤

J(u) +

K1 2

Z

(−γ∇2 u + 2vˆ∗0 u + 2(α − ε)u3 − f )2 dx, ∀u ∈ V1 . (11.64)

Ω

Therefore, from such results we may obtain Z K1 (−γ∇2 u + 2vˆ∗0 u + 2(α − ε)u3 − f )2 dx J(u0 ) = inf J(u) + u∈V1 2 Ω = J2∗ (u0 , vˆ∗0 ) ∗ ∗ (11.65) = sup inf J2 (u, v0 ) . v∗0 ∈B∗

u∈V1

The proof is complete.

Chapter 12

Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations

12.1

Introduction

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis, and the duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. In the first sections, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section we present some numerical results concerning the generalized method of lines applied to a Ginzburg-Landau type equation. Such results are based on the works of J.J. Telega and W.R. Bielski [10, 11, 74, 75] and on a D.C. optimization approach developed in Toland [81].

254

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The Method of Lines and Duality Principles for Non-Convex Models

Remark 12.1.1 This chapter has been published in a similar article format by the MDPI Journal Mathematics, reference [20]: F.S. Botelho, Dual Variational Formulations for a Large Class of NonConvex Models in the Calculus of Variations, Mathematics 2023, 11(1), 63; https://doi.org/10.3390/math11010063 - 24 Dec 2022. About the other references, details on the Sobolev spaces involved are found in [1]. Related results on convex analysis and duality theory are addressed in [12, 13, 14, 22, 62]. Finally, similar models on the superconductivity physics may be found in [4, 52]. Remark 12.1.2 It is worth highlighting, we may generically denote Z

[(−γ∇2 + KId )−1 v∗ ]v∗ dx

Ω

simply by (v∗ )2 dx, 2 Ω −γ∇ + K

Z

where Id denotes a concerning identity operator. Other similar notations may be used along this text as their indicated meaning are sufficiently clear. Also, ∇2 denotes the Laplace operator and for real constants K2 > 0 and K1 > 0, the notation K2 ≫ K1 means that K2 > 0 is much larger than K1 > 0. Finally, we adopt the standard Einstein convention of summing up repeated indices, unless otherwise indicated. In order to clarify the notation, here we introduce the definition of topological dual space. Definition 12.1.3 (Topological dual spaces) Let U be a Banach space. We shall define its dual topological space, as the set of all linear continuous functionals defined on U. We suppose such a dual space of U, may be represented by another Banach space U ∗ , through a bilinear form ⟨·, ·⟩U : U ×U ∗ → R (here we are referring to standard representations of dual spaces of Sobolev and Lebesgue spaces). Thus, given f : U → R linear and continuous, we assume the existence of a unique u∗ ∈ U ∗ such that f (u) = ⟨u, u∗ ⟩U , ∀u ∈ U.

(12.1)

The norm of f , denoted by ∥ f ∥U ∗ , is defined as ∥ f ∥U ∗ = sup{|⟨u, u∗ ⟩U | : ∥u∥U ≤ 1} ≡ ∥u∗ ∥U ∗ .

(12.2)

u∈U

At this point we start to describe the primal and dual variational formulations. Let Ω ⊂ R3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω.

Dual Variational Formulation for a Large Class of Non-Convex Models

■

255

Firstly we emphasize that, for the Banach space Y = Y ∗ = L2 (Ω), we have ⟨v, v∗ ⟩L2 =

Z

v v∗ dx, ∀v, v∗ ∈ L2 (Ω).

Ω

For the primal formulation we consider the functional J : U → R where γ 2

J(u) =

+

Z

∇u · ∇u dx

Ω

α 2

Z Ω

(u2 − β )2 dx − ⟨u, f ⟩L2 .

(12.3)

Here we assume α > 0, β > 0, γ > 0, U = W01,2 (Ω), f ∈ L2 (Ω). Moreover we denote Y = Y ∗ = L2 (Ω). Define also G1 : U → R by G1 (u) =

γ 2

Z

∇u · ∇u dx,

Ω

G2 : U ×Y → R by G2 (u, v) =

α 2

Z

(u2 − β + v)2 dx +

Ω

K 2

Z

u2 dx,

Ω

and F : U → R by K F(u) = 2

Z

u2 dx,

Ω

where K ≫ γ. It is worth highlighting that in such a case J(u) = G1 (u) + G2 (u, 0) − F(u) − ⟨u, f ⟩L2 , ∀u ∈ U. Furthermore, define the following specific polar functionals specified, namely, G∗1 : [Y ∗ ]2 → R by G∗1 (v∗1 + z∗ ) =

sup {⟨u, v∗1 + z∗ ⟩L2 − G1 (u)}

u∈U

1 2

=

Z

[(−γ∇2 )−1 (v∗1 + z∗ )](v∗1 + z∗ ) dx,

(12.4)

Ω

G∗2 : [Y ∗ ]2 → R by G∗2 (v∗2 , v∗0 ) =

sup

{⟨u, v∗2 ⟩L2 + ⟨v, v∗0 ⟩L2 − G2 (u, v)}

(u,v)∈U×Y

=

(v∗2 )2 dx ∗ Ω 2v0 + K Z Z 1 + (v∗0 )2 dx + β v∗0 dx, 2α Ω Ω 1 2

Z

(12.5)

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The Method of Lines and Duality Principles for Non-Convex Models

if v∗0 ∈ B∗ where

B∗ = {v∗0 ∈ Y ∗ : 2v∗0 + K > K/2 in Ω},

and finally, F ∗ : Y ∗ → R by sup {⟨u, z∗ ⟩L2 − F(u)}

F ∗ (z∗ ) =

u∈U

1 2K

=

Z

(z∗ )2 dx.

(12.6)

Ω

Define also A∗ = {v∗ = (v∗1 , v∗2 , v∗0 ) ∈ [Y ∗ ]2 × B∗ : v∗1 + v∗2 − f = 0, in Ω}, J ∗ : [Y ∗ ]4 → R by J ∗ (v∗ , z∗ ) = −G∗1 (v∗1 + z∗ ) − G∗2 (v∗2 , v∗0 ) + F ∗ (z∗ ) and J1∗ : [Y ∗ ]4 ×U → R by J1∗ (v∗ , z∗ , u) = J ∗ (v∗ , z∗ ) + ⟨u, v∗1 + v∗2 − f ⟩L2 .

12.2

The main duality principle, a convex dual formulation and the concerning proximal primal functional

Our main result is summarized by the following theorem. Theorem 12.2.1 Considering the definitions and statements in the last section, suppose also (vˆ∗ , zˆ∗ , u0 ) ∈ [Y ∗ ]2 × B∗ ×Y ∗ ×U is such that δ J1∗ (vˆ∗ , zˆ∗ , u0 ) = 0. Under such hypotheses, we have δ J(u0 ) = 0, vˆ∗ ∈ A∗ and Z K 2 J(u0 ) = inf J(u) + |u − u0 | dx u∈U 2 Ω = J ∗ (vˆ∗ , zˆ∗ ) = sup {J ∗ (v∗ , zˆ∗ )} . v∗ ∈A∗

(12.7)

Dual Variational Formulation for a Large Class of Non-Convex Models

Proof 12.1

■

257

Since δ J1∗ (vˆ∗ , zˆ∗ , u0 ) = 0

from the variation in v∗1 we obtain −

(vˆ∗1 + zˆ∗ ) + u0 = 0 in Ω, −γ∇2

so that vˆ∗1 + zˆ∗ = −γ∇2 u0 . From the variation in v∗2 we obtain −

vˆ∗2 + u0 = 0, in Ω. 2vˆ∗0 + K

From the variation in v∗0 we also obtain vˆ∗0 (vˆ∗2 )2 − −β = 0 (2vˆ∗0 + K)2 α and therefore, vˆ∗0 = α(u20 − β ). From the variation in u we get vˆ∗1 + vˆ∗2 − f = 0, in Ω and thus vˆ∗ ∈ A∗ . Finally, from the variation in z∗ , we obtain −

(vˆ∗1 + zˆ∗ ) zˆ∗ + = 0, in Ω. −γ∇2 K

so that −u0 +

zˆ∗ = 0, K

that is, zˆ∗ = Ku0 in Ω. From such results and vˆ∗ ∈ A∗ we get 0 = vˆ∗1 + vˆ∗2 − f = −γ∇2 u0 − zˆ∗ + 2(v∗0 )u0 + Ku0 − f = −γ∇2 u0 + 2α(u20 − β )u0 − f , so that δ J(u0 ) = 0.

(12.8)

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The Method of Lines and Duality Principles for Non-Convex Models

Also from this and from the Legendre transform proprieties we have G∗1 (vˆ∗1 + zˆ∗ ) = ⟨u0 , vˆ∗1 + zˆ∗ ⟩L2 − G1 (u0 ), G∗2 (vˆ∗2 , vˆ∗0 ) = ⟨u0 , vˆ∗2 ⟩L2 + ⟨0, v∗0 ⟩L2 − G2 (u0 , 0), F ∗ (ˆz∗ ) = ⟨u0 , zˆ∗ ⟩L2 − F(u0 ) and thus we obtain J ∗ (vˆ∗ , zˆ∗ ) = = = =

−G∗1 (vˆ∗1 + zˆ∗ ) − G∗2 (vˆ∗2 , vˆ∗0 ) + F ∗ (ˆz∗ ) −⟨u0 , vˆ∗1 + vˆ∗2 ⟩ + G1 (u0 ) + G2 (u0 , 0) − F(u0 ) −⟨u0 , f ⟩L2 + G1 (u0 ) + G2 (u0 , 0) − F(u0 ) J(u0 ).

(12.9)

Summarizing, we have got J ∗ (vˆ∗ , zˆ∗ ) = J(u0 ).

(12.10)

On the other hand J ∗ (vˆ∗ , zˆ∗ ) = ≤ = =

−G∗1 (vˆ∗1 + zˆ∗ ) − G∗2 (vˆ∗2 , vˆ∗0 ) + F ∗ (ˆz∗ ) −⟨u, vˆ∗1 + zˆ∗ ⟩L2 − ⟨u, vˆ∗2 ⟩L2 − ⟨0, v∗0 ⟩L2 + G1 (u) + G2 (u, 0) + F ∗ (ˆz∗ ) −⟨u, f ⟩L2 + G1 (u) + G2 (u, 0) − ⟨u, zˆ∗ ⟩L2 + F ∗ (ˆz∗ ) −⟨u, f ⟩L2 + G1 (u) + G2 (u, 0) − F(u) + F(u) − ⟨u, zˆ∗ ⟩L2 + F ∗ (ˆz∗ ) Z K u2 dx − ⟨u, zˆ∗ ⟩L2 + F ∗ (ˆz∗ ) = J(u) + 2 Ω Z Z K K u2 dx − K⟨u, u0 ⟩L2 + u2 dx = J(u) + 2 Ω 2 Ω 0 Z K |u − u0 |2 dx, ∀u ∈ U. (12.11) = J(u) + 2 Ω

Finally by a simple computation we may obtain the Hessian 2 ∗ ∗ ∗ ∂ J (v , z ) 0. Denoting v∗ = (v∗1 , v∗2 ), define J ∗ : U ×Y ∗ ×Y ∗ → R by 1 1 1 J ∗ (u, v∗ ) = ∥v∗1 − G′ (u)∥22 + ∥v∗2 − F ′ (u)∥22 + ∥v∗1 − v∗2 ∥22 2 2 2

(12.13)

Denoting L1∗ (u, v∗ ) = v∗1 − G′ (u) and L2∗ (u, v∗ ) = v∗2 − F ′ (u), define also 1 1 , C∗ = (u, v∗ ) ∈ U ×Y ∗ ×Y ∗ : ∥L1∗ (u, v∗1 )∥∞ ≤ and ∥L2∗ (u, v∗1 )∥∞ ≤ K K for an appropriate K > 0 to be specified. Observe that in C∗ the Hessian of J ∗ is given by ′′ 2 G (u) + F ′′ (u)2 + O(1/K) −G′′ (u) −F ′′ (u) −G′′ (u) 2 −1 , (12.14) {δ 2 J ∗ (u, v∗ )} = −F ′′ (u) −1 2 Observe also that

det

∂ 2 J ∗ (u, v∗ ) ∂ v∗1 ∂ v∗2

= 3,

and det{δ 2 J ∗ (u, v∗ )} = (G′′ (u) − F ′′ (u))2 + O(1/K) = (δ 2 J(u))2 + O(1/K). Define now vˆ∗1 = G′ (u0 ), vˆ∗2 = F ′ (u0 ), so that vˆ∗1 − vˆ∗2 = 0. From this we may infer that (u0 , vˆ∗1 , vˆ∗2 ) ∈ C∗ and J ∗ (u0 , vˆ∗ ) = 0 =

min J ∗ (u, v∗ ).

(u,v∗ )∈C∗

Moreover, for K > 0 sufficiently big, J ∗ is convex in a neighborhood of (u0 , vˆ∗ ). Therefore, in the last lines, we have proven the following theorem.

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The Method of Lines and Duality Principles for Non-Convex Models

Theorem 12.3.1 Under the statements and definitions of the last lines, there exist r0 > 0 and r1 > 0 such that J(u0 ) =

min J(u)

u∈Br0 (u0 )

and (u0 , vˆ∗1 , vˆ∗2 ) ∈ C∗ is such that J ∗ (u0 , vˆ∗ ) = 0 =

min

(u,v∗ )∈U×[Y ∗ ]2

J ∗ (u, v∗ ).

Moreover, J ∗ is convex in Br1 (u0 , vˆ∗ ).

12.4

One more duality principle and a concerning primal dual variational formulation

In this section, we establish a new duality principle and a related primal dual formulation. The results are based on the approach of Toland [81].

12.4.1

Introduction

Let Ω ⊂ R3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω. Let J : V → R be a functional such that J(u) = G(u) − F(u), ∀u ∈ V, where V = W01,2 (Ω). Suppose G, F are both three times Fr´echet differentiable convex functionals such that ∂ 2 G(u) >0 ∂ u2 and ∂ 2 F(u) >0 ∂ u2 ∀u ∈ V. Assume also there exists α1 ∈ R such that α1 = inf J(u). u∈V

Moreover, suppose that if {un } ⊂ V is such that ∥un ∥V → ∞

Dual Variational Formulation for a Large Class of Non-Convex Models

■

261

then J(un ) → +∞, as n → ∞. At this point we define J ∗∗ : V → R by J ∗∗ (u) =

sup

{⟨u, v∗ ⟩ + α},

(v∗ ,α)∈H ∗

where H ∗ = {(v∗ , α) ∈ V ∗ × R : ⟨v, v∗ ⟩V + α ≤ F(v), ∀v ∈ V }. Observe that (0, α1 ) ∈ H ∗ , so that J ∗∗ (u) ≥ α1 = inf J(u). u∈V

On the other hand, clearly we have J ∗∗ (u) ≤ J(u), ∀u ∈ V, so that we have got α1 = inf J(u) = inf J ∗∗ (u). u∈V

u∈V

Let u ∈ V . Since J is strongly continuous, there exist δ > 0 and A > 0 such that, α1 ≤ J ∗∗ (v) ≤ J(v) ≤ A, ∀v ∈ Bδ (u). From this, considering that J ∗∗ is convex on V , we may infer that J ∗∗ is continuous at u, ∀u ∈ V. Hence J ∗∗ is strongly lower semi-continuous on V , and since J ∗∗ is convex we may infer that J ∗∗ is weakly lower semi-continuous on V . Let {un } ⊂ V be a sequence such that 1 α1 ≤ J(un ) < α1 + , ∀n ∈ N. n Hence α1 = lim J(un ) = inf J(u) = inf J ∗∗ (u). n→∞

u∈V

u∈V

Suppose there exists a subsequence {unk } of {un } such that ∥unk ∥V → ∞, as k → ∞. From the hypothesis we have J(unk ) → +∞, as k → ∞, which contradicts α1 ∈ R.

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The Method of Lines and Duality Principles for Non-Convex Models

Therefore there exists K > 0 such that ∥un ∥V ≤ K, ∀u ∈ V. Since V is reflexive, from this and the Katutani Theorem, there exists a subsequence {unk } of {un } and u0 ∈ V such that unk ⇀ u0 , weakly in V. Consequently, from this and considering that J ∗∗ is weakly lower semicontinuous, we have got α1 = lim inf J ∗∗ (unk ) ≥ J ∗∗ (u0 ), k→∞

so that J ∗∗ (u0 ) = min J ∗∗ (u). u∈V

Define

G∗ , F ∗

: V∗

→ R by G∗ (v∗ ) = sup{⟨u, v∗ ⟩V − G(u)}, u∈V

and F ∗ (v∗ ) = sup{⟨u, v∗ ⟩V − F(u)}. u∈V

Defining also

J∗

: V → R by J ∗ (v∗ ) = F ∗ (v∗ ) − G∗ (v∗ ),

from the results in [81], we may obtain inf J(u) = ∗inf ∗ J ∗ (v∗ ), v ∈V

u∈V

so that J ∗∗ (u0 ) = =

inf J ∗∗ (u)

u∈V

inf J(u) = ∗inf ∗ J ∗ (v∗ ). v ∈V

u∈V

Suppose now there exists uˆ ∈ V such that J(u) ˆ = inf J(u). u∈V

From the standard necessary conditions, we have δ J(u) ˆ = 0,

(12.15)

Dual Variational Formulation for a Large Class of Non-Convex Models

so that

■

263

∂ G(u) ˆ ∂ F(u) ˆ − = 0. ∂u ∂u

Define now

∂ F(u) ˆ . ∂u From these last two equations we obtain v∗0 =

v∗0 =

∂ G(u) ˆ . ∂u

From such results and the Legendre transform properties, we have

so that δ J ∗ (v∗0 ) =

uˆ =

∂ F ∗ (v∗0 ) , ∂ v∗

uˆ =

∂ G∗ (v∗0 ) , ∂ v∗

∂ F ∗ (v∗0 ) ∂ G∗ (v∗0 ) − = uˆ − uˆ = 0, ∂ v∗ ∂ v∗

G∗ (v∗0 ) = ⟨u, ˆ v∗0 ⟩V − G(u) ˆ and F ∗ (v∗0 ) = ⟨u, ˆ v∗0 ⟩V − F(u) ˆ so that inf J(u) =

u∈V

J(u) ˆ

= G(u) ˆ − F(u) ˆ ∗ ∗ = ∗inf ∗ J (v ) v ∈V

= F ∗ (v∗0 ) − G∗ (v∗0 ) = J ∗ (v∗0 ).

12.4.2

(12.16)

The main duality principle and a related primal dual variational formulation

Considering these last statements and results, we may prove the following theorem. Theorem 12.4.1 Let Ω ⊂ R3 be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by ∂ Ω.

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The Method of Lines and Duality Principles for Non-Convex Models

Let J : V → R be a functional such that J(u) = G(u) − F(u), ∀u ∈ V, where V = W01,2 (Ω). Suppose G, F are both three times Fr´echet differentiable functionals such that there exists K > 0 such that ∂ 2 G(u) +K > 0 ∂ u2 and ∂ 2 F(u) +K > 0 ∂ u2 ∀u ∈ V. Assume also there exists u0 ∈ V and α1 ∈ R such that α1 = inf J(u) = J(u0 ). u∈V

Assume K3 > 0 is such that ∥u0 ∥∞ < K3 . Define V˜ = {u ∈ V : ∥u∥∞ ≤ K3 }. Assume K1 > 0 is such that if u ∈ V˜ then max ∥F ′ (u)∥∞ , ∥G′ (u)∥∞ , ∥F ′′ (u)∥∞ , ∥F ′′′ (u)∥∞ , ∥G′′ (u)∥∞ , ∥G′′′ (u)∥∞ ≤ K1 . Suppose also K ≫ max{K1 , K3 }. Define FK , GK : V → R by FK (u) = F(u) +

K 2

Z

K 2

Z

u2 dx,

Ω

and GK (u) = G(u) +

u2 dx,

Ω

∀u ∈ V. Define also G∗K , FK∗ : V ∗ → R by G∗K (v∗ ) = sup{⟨u, v∗ ⟩V − GK (u)}, u∈V

and FK∗ (v∗ ) = sup{⟨u, v∗ ⟩V − FK (u)}. u∈V

Dual Variational Formulation for a Large Class of Non-Convex Models

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265

Observe that since u0 ∈ V is such that J(u0 ) = inf J(u), u∈V

we have δ J(u0 ) = 0. Let ε > 0 be a small constant. Define ∂ FK (u0 ) ∈ V ∗. ∂u Under such hypotheses, defining J1∗ : V ×V ∗ → R by v∗0 =

J1∗ (u, v∗ ) =

FK∗ (v∗ ) − G∗K (v∗ )

2

2

∗ ∗

∗ ∗

1 1 ∂ GK (v ) ∂ FK (v )

+ + − u − u

2ε ∂ v∗ 2ε ∂ v∗ 2 2

∗ ∗ ∗ (v∗ ) 2 1 ∂ F ∂ G (v ) K + − K∗ , 2ε ∂ v∗ ∂ v 2

(12.17)

we have J(u0 ) = = =

inf J(u)

u∈V

inf

(u,v∗ )∈V ×V ∗ J1∗ (u0 , v∗0 ).

J1∗ (u, v∗ ) (12.18)

Proof 12.2 Observe that from the hypotheses and the results and statements of the last subsection J(u0 ) = inf J(u) = ∗inf ∗ JK∗ (v∗ ) = JK∗ (v∗0 ), u∈V

v ∈Y

where JK∗ (v∗ ) = FK∗ (v∗ ) − G∗K (v∗ ), ∀v∗ ∈ V ∗ . Moreover we have J1∗ (u, v∗ ) ≥ JK∗ (v∗ ), ∀u ∈ V, v∗ ∈ V ∗ . Also from hypotheses and the last subsection results, u0 =

∂ FK∗ (v∗0 ) ∂ G∗K (v∗0 ) = , ∂ v∗ ∂ v∗

so that clearly we have J1∗ (u0 , v∗0 ) = JK∗ (v∗0 ).

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The Method of Lines and Duality Principles for Non-Convex Models

From these last results, we may infer that J(u0 ) = = = = =

inf J(u)

u∈V

inf JK∗ (v∗ ) v∗ ∈V ∗ JK∗ (v∗0 ) inf

(u,v∗ )∈V ×V ∗ J1∗ (u0 , v∗0 ).

J1∗ (u, v∗ ) (12.19)

The proof is complete. Remark 12.4.2 At this point we highlight that J1∗ has a large region of convexity around the optimal point (u0 , v∗0 ), for K > 0 sufficiently large and corresponding ε > 0 sufficiently small. Indeed, observe that for v∗ ∈ V ∗ , ˆ v∗ ⟩V − GK (u) ˆ G∗K (v∗ ) = sup{⟨u, v∗ ⟩V − GK (u)} = ⟨u, u∈V

where uˆ ∈ V is such that v∗ =

∂ GK (u) ˆ = G′ (u) ˆ + K u. ˆ ∂u

Taking the variation in v∗ in this last equation, we obtain 1 = G′′ (u)

∂ uˆ ∂ uˆ +K ∗, ∗ ∂v ∂v

so that 1 ∂ uˆ = ′′ =O ∂ v∗ G (u) + K

1 . K

From this we get ∂ 2 uˆ ∂ (v∗ )2

= −

1 (G′′ (u) + K)2

G′′′ (u)

∂ uˆ ∂ v∗

1 G′′′ (u) (G′′ (u) + K)3 1 . = O K3

= −

On the other hand, from the implicit function theorem ∂ G∗K (v∗ ) ∂ uˆ = u + [v∗ − G′K (u)] ˆ = u, ∂ v∗ ∂ v∗ so that

∂ uˆ ∂ 2 G∗K (v∗ ) = ∗ =O ∗ 2 ∂ (v ) ∂v

1 K

(12.20)

Dual Variational Formulation for a Large Class of Non-Convex Models

and

∂ 3 G∗K (v∗ ) ∂ 2 uˆ = =O ∂ (v∗ )3 ∂ (v∗ )2

1 K3

■

267

.

Similarly, we may obtain ∂ 2 FK∗ (v∗ ) =O ∂ (v∗ )2 and

∂ 3 FK∗ (v∗ ) =O ∂ (v∗ )3

1 K

1 K3

.

Denoting A=

∂ 2 FK∗ (v∗0 ) ∂ (v∗ )2

B=

∂ 2 G∗K (v∗0 ) , ∂ (v∗ )2

and

we have ∂ 2 J1∗ (u0 , v∗0 ) 1 = A − B + 2A2 + 2B2 − 2AB , ∂ (v∗ )2 ε ∂ 2 J1∗ (u0 , v∗0 ) 2 = , ∂ u2 ε and

∂ 2 J1∗ (u0 , v∗0 ) 1 = − (A + B). ∗ ∂ (v )∂ u ε

From this we get det(δ 2 J ∗ (v∗0 , u0 ))

2 2 ∗ ∂ 2 J1∗ (u0 , v∗0 ) ∂ 2 J1∗ (u0 , v∗0 ) ∂ J1 (u0 , v∗0 ) − ∂ (v∗ )2 ∂ u2 ∂ (v∗ )∂ u

=

A−B (A − B)2 +2 ε2 ε 1 = O ε2 ≫ 0 =

2

about the optimal point (u0 , v∗0 ).

(12.21)

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12.5

The Method of Lines and Duality Principles for Non-Convex Models

One more dual variational formulation

In this section, again for Ω ⊂ R3 an open, bounded, connected set with a regular (Lipschitzian) boundary ∂ Ω, γ > 0, α > 0, β > 0 and f ∈ L2 (Ω), we denote F1 : V ×Y → R, F2 : V → R and G : V ×Y → R by γ 2

F1 (u, v∗0 ) =

+

Z

K 2

∇u · ∇u dx −

Ω

K1 2

Z

u2 dx

Ω

(−γ∇2 u + 2v∗0 u − f )2 dx +

Ω

F2 (u) = and G(u, v) =

Z

α 2

Z

K2 2

Z Ω

K2 2

Z

u2 dx,

(12.22)

Ω

u2 dx + ⟨u, f ⟩L2 ,

(u2 − β + v)2 dx +

Ω

K 2

Z

u2 dx.

Ω

We define also J1 (u, v∗0 ) = F1 (u, v∗0 ) − F2 (u) + G(u, 0), J(u) =

γ 2

Z

∇u · ∇u dx +

Ω

Z

α 2

Ω

(u2 − β )2 dx − ⟨u, f ⟩L2 ,

and F1∗ : [Y ∗ ]3 → R, F2∗ : Y ∗ → R, and G∗ : [Y ∗ ]2 → R, by F1∗ (v∗2 , v∗1 , v∗0 ) = sup{⟨u, v∗1 + v∗2 ⟩L2 − F1 (u, v∗0 )} u∈V

=

2 v∗1 + v∗2 + K1 (−γ∇2 + 2v∗0 ) f ∗ 2 dx 2 2 Ω (−γ∇ − K + K2 + K1 (−γ∇ + 2v0 ) ) Z K1 − f 2 dx, 2 Ω 1 2

Z

F2∗ (v∗2 ) =

(12.23)

sup{⟨u, v∗2 ⟩L2 − F2 (u)} u∈V

1 2K2

=

Z

(v∗2 − f )2 dx,

(12.24)

Ω

and G∗ (v∗1 , v∗0 ) = =

sup (u,v)∈V ×Y

1 2

{⟨u, v∗1 ⟩L2 − ⟨v, v∗0 ⟩L2 − G(u, v)}

(v∗1 )2 1 ∗ + K dx + 2α 2v Ω 0

Z

Z

+β Ω

v∗0 dx

Z

(v∗0 )2 dx

Ω

(12.25)

Dual Variational Formulation for a Large Class of Non-Convex Models

if v∗0 ∈ B∗ where

■

269

B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K/2}.

Define also V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, +

A = {u ∈ V : u f ≥ 0 a.e. in Ω}, V1 = V2 ∩ A+ , B∗2 = {v∗0 ∈ Y ∗ : −γ∇2 − K + K1 (−γ∇2 + 2v∗0 )2 > 0},

D∗3 = {(v∗1 , v∗2 ) ∈ Y ∗ ×Y ∗ : −1/α + 4K1 [u(v∗1 , v∗2 , v∗0 )2 ] + 100/K2 ≤ 0, ∀v∗0 ∈ B∗ }, where u(v∗2 , v∗0 ) =

ϕ1 , ϕ

ϕ1 = (v∗1 + v∗2 + K1 (−γ∇2 + 2v∗0 ) f ) and ϕ = (−γ∇2 − K + K1 (−γ∇2 + 2v∗0 )2 + K2 ), D∗ = {v∗2 ∈ Y ∗ ; ∥v∗2 ∥∞ < K4 } E ∗ = {v∗1 ∈ Y ∗ : ∥v∗1 ∥∞ ≤ K5 }, for some K3 , K4 , K5 > 0 to be specified, Finally, we also define J1∗ : [Y ∗ ]2 × B∗ → R, J1∗ (v∗2 , v∗1 , v∗0 ) = −F1∗ (v∗2 , v∗1 , v∗0 ) + F2∗ (v∗2 ) − G∗ (v∗1 , v∗0 ). Assume now K1 = 1/[4(α + ε)K32 ], K2 ≫ K1 ≫ max{K3 , K4 , K5 , 1, γ, α, β }. Observe that, by direct computation, we may obtain ∂ 2 J1∗ (v∗2 , v∗1 , v∗0 ) 1 = − + 4K1 u(v∗ )2 + O(1/K2 ) < 0, ∗ 2 ∂ (v0 ) α for v∗0 ∈ B∗3 . Considering such statements and definitions, we may prove the following theorem.

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The Method of Lines and Duality Principles for Non-Convex Models

Theorem 12.5.1 Let (vˆ∗2 , vˆ∗1 , vˆ∗0 ) ∈ ((D∗ × E ∗ ) ∩ D∗3 ) × (B∗2 ∩ B∗ ) be such that δ J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = 0 and u0 ∈ V1 , where u0 =

vˆ∗1 + vˆ∗2 + K1 (−γ∇2 + 2v∗0 ) f . K2 − K − γ∇2 + K1 (−γ∇2 + 2vˆ∗0 )2

Under such hypotheses, we have δ J(u0 ) = 0, so that J(u0 ) = =

Z K1 (−γ∇2 u + 2vˆ∗0 u − f )2 dx J(u) + u∈V1 2 Ω ) ( inf

inf

v∗2 ∈D∗

sup (v∗1 ,v∗0 )∈E ∗ ×B∗

J1∗ (v∗2 , v∗1 , v∗0 )

= J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ).

(12.26)

Proof 12.3 Observe that δ J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = 0 so that, since (vˆ∗2 , vˆ∗1 ) ∈ D∗3 ,vˆ∗0 ∈ B∗2 and J1∗ is quadratic in v∗2 , we may infer that J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) =

inf J1∗ (v∗2 , vˆ∗1 , vˆ∗0 )

v∗2 ∈Y ∗

=

sup (v∗1 ,v∗0 )∈E ∗ ×B∗

J1∗ (vˆ∗2 , v∗1 , v∗0 ).

Therefore, from a standard saddle point theorem, we have that ) ( J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) ∗inf ∗ v2 ∈Y

sup (v∗1 ,v∗0 )∈E ∗ ×B∗

J1∗ (v∗2 , v∗1 , v∗0 ) .

Now we are going to show that δ J(u0 ) = 0. From

∂ J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = 0, ∂ v∗2

we have −u0 +

vˆ∗2 = 0, K2

and thus vˆ∗2 = K2 u0 .

(12.27)

Dual Variational Formulation for a Large Class of Non-Convex Models

From

■

271

∂ J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = 0, ∂ v∗1

we obtain −u0 −

vˆ∗1 − f = 0, 2vˆ∗0 + K

and thus vˆ∗1 = −2vˆ∗0 u0 − Ku0 + f . Finally, denoting D = −γ∇2 u0 + 2vˆ∗0 u0 − f , from

∂ J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = 0, ∂ v∗0

we have −2Du0 + u20 −

vˆ∗0 − β = 0, α

so that vˆ∗0 = α(u20 − β − 2Du0 ).

(12.28)

Observe now that vˆ∗1 + vˆ∗2 + K1 (−γ∇2 + 2vˆ∗0 ) f = (K2 − K − γ∇2 + K1 (−γ∇2 + 2vˆ∗0 )2 )u0 so that K2 u0 − 2vˆ0 u0 − Ku0 + f = K2 u0 − Ku0 − γ∇2 u0 + K1 (−γ∇2 + 2vˆ∗0 )(−γ∇2 u0 + 2vˆ∗0 u0 − f ). (12.29) The solution for this last system of equations (12.28) and (12.29) is obtained through the relations vˆ∗0 = α(u20 − β ) and −γ∇2 u0 + 2vˆ∗0 u0 − f = D = 0, so that δ J(u0 ) = −γ∇2 u0 + 2α(u20 − β )u0 − f = 0 and

Z K1 (−γ∇2 u0 + 2vˆ∗0 u0 − f )2 dx = 0. δ J(u0 ) + 2 Ω Moreover, from the Legendre transform properties F1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = ⟨u0 , vˆ∗2 + vˆ∗1 ⟩L2 − F1 (u0 , vˆ∗0 ), F2∗ (vˆ∗2 ) = ⟨u0 , vˆ∗2 ⟩L2 − F2 (u0 ),

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G∗ (vˆ∗1 , vˆ∗0 ) = −⟨u0 , vˆ∗1 ⟩L2 − ⟨0, vˆ∗0 ⟩L2 − G(u0 , 0), so that J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) = −F1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) + F2∗ (vˆ∗2 ) − G∗ (vˆ∗1 , vˆ∗0 ) = F1 (u0 , vˆ∗0 ) − F2 (u0 ) + G(u0 , 0) = J(u0 ).

(12.30)

Observe now that J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ) Z Z γ K ∇u · ∇u dx − u2 dx ≤ 2 Ω 2 Ω Z K1 + (−γ∇2 u + vˆ∗0 u − f )2 dx + ⟨u, vˆ∗1 ⟨L2 −⟨u, f ⟩L2 2 Ω Z Z Z (v∗1 )2 1 1 ∗ 2 − dx − (v ) dx − β v∗0 dx 0 2 Ω 2v∗0 + K 2α Ω Ω Z γ ∇u · ∇u dx − ⟨u, f ⟩L2 ≤ 2 Ω Z K1 (−γ∇2 u + vˆ∗0 u − f )2 dx + 2 Ω Z (v∗1 )2 1 ∗ + sup +⟨u, vˆ1 ⟨L2 − dx 2 Ω 2v∗0 + K (v∗1 ,v∗0 )∈D∗ ×B∗ Z Z Z 1 1 − (v∗0 )2 dx − (v∗0 )2 dx − β v∗0 dx 2α Ω 2α Ω Ω Z K1 (−γ∇2 u + 2vˆ∗0 u − f )2 dx, = J(u) + 2 Ω

J(u0 ) =

(12.31)

∀u ∈ V1 . Hence, we have got Z K1 J(u) + (−γ∇2 u + 2vˆ∗0 u − f )2 dx . u∈V1 2 Ω

J(u0 ) = inf

Joining the pieces, we have got Z K1 J(u0 ) = inf J(u) + (−γ∇2 u + 2vˆ∗0 u − f )2 dx u∈V 2 Ω ( ) =

inf

v∗2 ∈Y ∗

sup (v∗1 ,v∗0 )∈E ∗ ×(B∗ ∩Br (vˆ∗0 ))

= J1∗ (vˆ∗2 , vˆ∗1 , vˆ∗0 ). The proof is complete.

J1∗ (v∗2 , v∗1 , v∗0 ) (12.32)

Dual Variational Formulation for a Large Class of Non-Convex Models

12.6

■

273

Another dual variational formulation

In this section, again for Ω ⊂ R3 an open, bounded, connected set with a regular (Lipschitzian) boundary ∂ Ω, γ > 0, α > 0, β > 0 and f ∈ L2 (Ω), we denote F1 : V ×Y → R, F2 : V → R and G : Y → R by γ 2

F1 (u, v∗0 ) =

+

Z Ω

K1 2

∇u · ∇u dx + ⟨u2 , v∗0 ⟩L2 Z

(−γ∇2 u + 2v∗0 u − f )2 dx +

Ω

F2 (u) = and

K2 2

Z

α G(u ) = 2

Z

u2 dx,

(12.33)

Ω

u2 dx + ⟨u, f ⟩L2 ,

Ω

2

K2 2

Z

(u2 − β )2 dx.

Ω

We define also J1 (u, v∗0 ) = F1 (u, v∗0 ) − F2 (u) − ⟨u2 , v∗0 ⟩L2 + G(u2 ), J(u) =

γ 2

Z

α 2

∇u · ∇u dx +

Ω +

Z Ω

(u2 − β )2 dx − ⟨u, f ⟩L2 ,

A = {u ∈ V : u f > 0, a.e. in Ω}, V2 = {u ∈ V : ∥u∥∞ ≤ K3 }, V1 = A+ ∩V2 , and F1∗ : [Y ∗ ]2 → R, F2∗ : Y ∗ → R, and G∗ : Y ∗ → R, by F1∗ (v∗2 , v∗0 ) = sup{⟨u, v∗2 ⟩L2 − F1 (u, v∗0 )} u∈V

=

2 v∗2 + K1 (−γ∇2 + 2v∗0 ) f ∗ ∗ 2 dx 2 2 Ω (−γ∇ + 2v0 + K2 + K1 (−γ∇ + 2v0 ) ) Z K1 f 2 dx, − 2 Ω 1 2

Z

F2∗ (v∗2 ) = =

(12.34)

sup{⟨u, v∗2 ⟩L2 − F2 (u)} u∈V

1 2K2

Z

(v∗2 + f )2 dx,

(12.35)

Ω

and G∗ (v∗0 ) = =

sup{⟨v, v∗0 ⟩L2 − G(v)} v∈Y

1 2α

Z Ω

(v∗0 )2 dx + β

Z Ω

v∗0 dx

(12.36)

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The Method of Lines and Duality Principles for Non-Convex Models

At this point we define B∗ = {v∗0 ∈ Y ∗ : ∥v∗0 ∥∞ ≤ K/2}, B∗2 = {v∗0 ∈ Y ∗ : −γ∇2 + 2v∗0 + K1 (−γ∇2 + 2v∗0 )2 > 0}, D∗3 = {v∗2 ∈ Y ∗ : −1/α + 4K1 [u(v∗2 , v∗0 )2 ] + 100/K2 ≤ 0, ∀v∗0 ∈ B∗ }, where u(v∗2 , v∗0 ) =

ϕ1 , ϕ

ϕ1 = (v∗2 + K1 (−γ∇2 + 2v∗0 ) f ) and ϕ = (−γ∇2 + 2v∗0 + K1 (−γ∇2 + 2v∗0 )2 + K2 ), Finally, we also define E1∗ = {v∗2 ∈ Y ∗ : ∥v∗2 ∥∞ ≤ (5/4)K2 }. E2∗ = {v∗2 ∈ Y ∗ : f v∗2 > 0, a.e. in Ω}, E ∗ = E1∗ ∩ E2∗ , and J1∗ : E ∗ × B∗ → R, by J1∗ (v∗2 , v∗0 ) = −F1∗ (v∗2 , v∗0 ) + F2∗ (v∗2 ) − G∗ (v∗0 ). Moreover, assume K2 ≫ K1 ≫ K ≫ K3 ≫ max{1, γ, α}. By directly computing δ 2 J1∗ (v∗2 , v∗0 ), recalling that ϕ = (−γ∇2 + 2v∗0 + K1 (−γ∇2 + 2v∗0 )2 + K2 ), ϕ1 = (v∗2 + K1 (−γ∇2 + 2v∗0 ) f ), and u=

ϕ1 , ϕ

we obtain ∂ 2 J1∗ (v∗2 , v∗0 ) = 1/K2 − 1/ϕ, ∂ (v∗2 )2 and

∂ 2 J1∗ (v∗2 , v∗0 ) = 4u2 K1 − 1/α + O(1/K2 ) < 0, ∂ (v∗0 )2

in E ∗ × B∗ . Considering such statements and definitions, we may prove the following theorem.

Dual Variational Formulation for a Large Class of Non-Convex Models

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275

Theorem 12.6.1 Let (vˆ∗2 , vˆ∗0 ) ∈ (E ∗ ∩ D∗3 ) × (B∗ ∩ B∗2 ) be such that δ J1∗ (vˆ∗2 , vˆ∗0 ) = 0 and u0 ∈ V1 be such that u0 =

vˆ∗2 + K1 (−γ∇2 + 2vˆ∗0 ) f . K2 + 2vˆ∗0 − γ∇2 + K1 (−γ∇2 + 2vˆ∗0 )2

Under such hypotheses, we have δ J(u0 ) = 0, so that J(u0 ) = =

Z K1 (−γ∇2 u + 2vˆ∗0 u − f )2 dx J(u) + u∈V1 2 Ω ( ) inf

inf

v∗2 ∈E ∗

sup J1∗ (v∗2 , v∗0 )

v∗0 ∈B∗

= J1∗ (vˆ∗2 , vˆ∗0 ).

(12.37)

Proof 12.4 Observe that δ J1∗ (vˆ∗2 , vˆ∗0 ) = 0 so that, since vˆ∗2 ∈ D∗3 , vˆ∗0 ∈ B∗2 and J1∗ is quadratic in v∗2 , we get sup J1∗ (vˆ∗2 , v∗0 ) = J1∗ (vˆ∗2 , vˆ∗0 ) = ∗inf ∗ J1∗ (v∗2 , vˆ∗0 ). v2 ∈E

v∗0 ∈B∗

Consequently, from this and the Min-Max Theorem, we obtain ) ( J1∗ (vˆ∗2 , vˆ∗0 ) = ∗inf ∗ v2 ∈E

sup

v∗0 ∈B∗

J1∗ (v∗2 , v∗0 )

= sup

v∗0 ∈B∗

Now we are going to show that δ J(u0 ) = 0. From

∂ J1∗ (vˆ∗2 , vˆ∗0 ) = 0, ∂ v∗2

we have −u0 +

vˆ∗2 = 0, K2

and thus vˆ∗2 = K2 u0 . Finally, denoting D = −γ∇2 u0 + 2vˆ∗0 u0 − f ,

inf J1∗ (v∗2 , v∗0 ) v∗2 ∈E ∗

.

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from

∂ J1∗ (vˆ∗2 , vˆ∗0 ) = 0, ∂ v∗0

we have −2Du0 + u20 −

vˆ∗0 − β = 0, α

so that vˆ∗0 = α(u20 − β − 2Du0 ).

(12.38)

Observe now that vˆ∗2 + K1 (−γ∇2 + 2vˆ∗0 ) f = (K2 − γ∇2 + 2vˆ∗0 + K1 (−γ∇2 + 2vˆ∗0 )2 )u0 so that K2 u0 − 2vˆ0 u0 − Ku0 + f = K2 u0 − Ku0 − γ∇2 u0 + K1 (−γ∇2 + 2vˆ∗0 )(−γ∇2 u0 + 2vˆ∗0 u0 − f ). (12.39) The solution for this last equation is obtained through the relation −γ∇2 u0 + 2vˆ∗0 u0 − f = D = 0, so that from this and (12.38), we get vˆ∗0 = α(u20 − β ). Thus, δ J(u0 ) = −γ∇2 u0 + 2α(u20 − β )u0 − f = 0 and

Z K1 2 2 ∗ δ J(u0 ) + (−γ∇ u0 + 2vˆ0 u0 − f ) dx = 0. 2 Ω Moreover, from the Legendre transform properties F1∗ (vˆ∗2 , vˆ∗0 ) = ⟨u0 , vˆ∗2 ⟩L2 − F1 (u0 , vˆ∗0 ), F2∗ (vˆ∗2 ) = ⟨u0 , vˆ∗2 ⟩L2 − F2 (u0 ), G∗ (vˆ∗0 ) = ⟨u20 , vˆ∗0 ⟩L2 − G(u20 ),

so that J1∗ (vˆ∗2 , vˆ∗0 ) = −F1∗ (vˆ∗2 , vˆ∗0 ) + F2∗ (vˆ∗2 ) − G∗ (vˆ∗0 ) = F1 (u0 , vˆ∗0 ) − F2 (u0 ) − ⟨u20 , vˆ∗0 ⟩L2 + G(u20 ) = J(u0 ).

(12.40)

Dual Variational Formulation for a Large Class of Non-Convex Models

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277

Observe now that J1∗ (vˆ∗2 , vˆ∗0 ) Z γ ∇u · ∇u dx − ⟨u2 , vˆ∗0 ⟩L2 ≤ 2 Ω Z K1 + (−γ∇2 u + vˆ∗0 u − f )2 dx − ⟨u, f ⟩L2 2 Ω Z Z 1 − (v∗0 )2 dx − β v∗0 dx 2α Ω Ω Z γ ∇u · ∇u dx − ⟨u, f ⟩L2 ≤ 2 Ω Z K1 (−γ∇2 u + vˆ∗0 u − f )2 dx + 2 Ω Z Z 1 ∗ 2 2 ∗ ∗ (v ) dx − β v0 dx + sup −⟨u , vˆ0 ⟩L2 − 2α Ω 0 Ω v∗0 ∈Y ∗

J(u0 ) =

= J(u) +

K1 2

Z

(−γ∇2 u + 2vˆ∗0 u − f )2 dx.

(12.41)

Ω

∀u ∈ V1 . Hence, we have got Z K1 J(u) + (−γ∇2 u + 2vˆ∗0 u − f )2 dx . u∈V1 2 Ω

J(u0 ) = inf

Joining the pieces, we have got Z K1 2 ∗ 2 (−γ∇ u + 2vˆ0 u − f ) dx J(u0 ) = inf J(u) + u∈V1 2 Ω ) ( =

inf

v∗2 ∈E ∗

sup J1∗ (v∗2 , v∗0 )

v∗0 ∈B∗

= J1∗ (vˆ∗2 , vˆ∗0 ).

(12.42)

The proof is complete.

12.7

A related numerical computation through the generalized method of lines

We start by recalling that the generalized method of lines was originally introduced in the book entitled “Topics on Functional Analysis, Calculus of Variations and Duality” [22], published in 2011. Indeed, the present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [22, 17, 12, 13]. About the Sobolev spaces involved, we would mention [1]. Concerning the applications, related models in physics are addressed in [4, 52].

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We also emphasize that, in such a method, the domain of the partial differential equation in question is discretized in lines (or more generally, in curves), and the concerning solution is written on these lines as functions of boundary conditions and the domain boundary shape. In fact, in its previous format, this method consists of an application of a kind of a partial finite differences procedure combined with the Banach fixed point theorem to obtain the relation between two adjacent lines (or curves). In the present chapter, we propose an improvement concerning the way we truncate the series solution obtained through an application of the Banach fixed point theorem to find the relation between two adjacent lines. The results obtained are very good even as a typical parameter ε > 0 is very small. In the next lines and sections we develop in details such a numerical procedure.

12.7.1

About a concerning improvement for the generalized method of lines

Let Ω ⊂ R2 where Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}. Consider the problem of solving the partial differential equation 2 2 −ε ∂∂ r2u + 1r ∂∂ ur + r12 ∂∂ θu2 + αu3 − β u = f , in Ω, u = u0 (θ ), on ∂ Ω1 , u = u f (θ ), on ∂ Ω2 .

(12.43)

Here Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ε > 0, α > 0, β > 0, and f ≡ 1, on Ω. In a partial finite differences scheme, such a system stands for un+1 − 2un + un−1 1 un − un−1 1 ∂ 2 un + + + αu3n − β un = fn , −ε d2 tn d tn2 ∂ θ 2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0, and uN = 0. Here N is the number of lines and d = 1/N.

Dual Variational Formulation for a Large Class of Non-Convex Models

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279

In particular, for n = 1 we have u2 − 2u1 + u0 1 (u1 − u0 ) 1 ∂ 2 u1 −ε + + αu31 − β u1 = f1 , + 2 d2 t1 d t1 ∂ θ 2 so that 1 1 ∂ 2 u1 2 d2 3 /3.0, u1 = u2 + u1 + u0 + (u1 − u0 ) d + 2 d + (−αu + β u − f ) 1 1 1 t1 ε t1 ∂ θ 2 We solve this last equation through the Banach fixed point theorem, obtaining u1 as a function of u2 . Indeed, we may set u01 = u2 and uk+1 1

=

1 ∂ 2 uk1 2 1 d u2 + uk1 + u0 + (uk1 − u0 ) d + 2 t1 t1 ∂ θ 2 d2 k 3 k +(−α(u1 ) + β u1 − f1 ) /3.0, ε

(12.44)

∀k ∈ N. Thus, we may obtain u1 = lim uk1 ≡ H1 (u2 , u0 ). k→∞

Similarly, for n = 2, we have 1 ∂ 2 u2 2 1 d u2 = u3 + u2 + H1 (u2 , u0 ) + (u2 − H1 (u2 , u0 )) d + 2 t1 t1 ∂ θ 2 d2 +(−αu32 + β u2 − f2 ) /3.0, (12.45) ε We solve this last equation through the Banach fixed point theorem, obtaining u2 as a function of u3 and u0 . Indeed, we may set u02 = u3 and uk+1 2

∀k ∈ N.

1 ∂ 2 uk2 2 1 d = u3 + uk2 + H1 (uk2 , u0 ) + (uk2 − H1 (uk2 , u0 )) d + 2 t2 t2 ∂ θ 2 d2 +(−α(uk2 )3 + β uk2 − f2 ) /3.0, (12.46) ε

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The Method of Lines and Duality Principles for Non-Convex Models

Thus, we may obtain u2 = lim uk2 ≡ H2 (u3 , u0 ). k→∞

Now reasoning inductively, having un−1 = Hn−1 (un , u0 ), we may get un

=

1 ∂ 2 un 2 1 d un+1 + un + Hn−1 (un , u0 ) + (un − Hn−1 (un , u0 )) d + 2 tn tn ∂ θ 2 d2 3 +(−αun + β un − fn ) /3.0, (12.47) ε

We solve this last equation through the Banach fixed point theorem, obtaining un as a function of un+1 and u0 . Indeed, we may set u0n = un+1 and uk+1 n

=

1 1 ∂ 2 ukn 2 d un+1 + ukn + Hn−1 (ukn , u0 ) + (ukn − Hn−1 (ukn , u0 )) d + 2 tn tn ∂ θ 2 d2 k 3 k +(−α(un ) + β un − fn ) /3.0, (12.48) ε

∀k ∈ N. Thus, we may obtain un = lim ukn ≡ Hn (un+1 , u0 ). k→∞

We have obtained un = Hn (un+1 , u0 ), ∀n ∈ {1, · · · , N − 1}. In particular, uN = u f (θ ), so that we may obtain uN−1 = HN−1 (uN , u0 ) = HN−1 (0) ≡ FN−1 (uN , u0 ) = FN−1 (u f (θ ), u0 (θ )). Similarly, uN−2 = HN−2 (uN−1 , u0 ) = HN−2 (HN−1 (uN , u0 )) = FN−2 (uN , u0 ) = FN−1 (u f (θ ), u0 (θ )),

an so on, up to obtaining u1 = H1 (u2 ) ≡ F1 (uN , u0 ) = F1 (u f (θ ), u0 (θ )). The problem is then approximately solved.

Dual Variational Formulation for a Large Class of Non-Convex Models

12.7.2

■

281

Software in Mathematica for solving such an equation

We recall that the equation to be solved is a Ginzburg-Landau type one, where 2 2 −ε ∂∂ r2u + 1r ∂∂ ur + r12 ∂∂ θu2 + αu3 − β u = f , in Ω, (12.49) u = 0, on ∂ Ω1 , u = u f (θ ), on ∂ Ω2 . Here Ω = {(r, θ ) ∈ R2 : 1 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}, ∂ Ω1 = {(1, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ∂ Ω2 = {(2, θ ) ∈ R2 : 0 ≤ θ ≤ 2π}, ε > 0, α > 0, β > 0, and f ≡ 1, on Ω. In a partial finite differences scheme, such a system stands for un+1 − 2un + un−1 1 un − un−1 1 ∂ 2 un + + + αu3n − β un = fn , −ε d2 tn d tn2 ∂ θ 2 ∀n ∈ {1, · · · , N − 1}, with the boundary conditions u0 = 0, and uN = u f [x]. Here N is the number of lines and d = 1/N. At this point, we present the concerning software for an approximate solution. Such a software is for N = 10 (10 lines) and u0 [x] = 0. ************************************* 1. m8 = 10; (N = 10 lines) 2. d = 1/m8; 3. e1 = 0.1; (ε = 0.1) 4. A = 1.0; 5. B = 1.0; 6. For[i = 1, i < m8, i + +, f [i] = 1.0]; ( f ≡ 1, on Ω) 7. a = 0.0; 8. For[i = 1, i < m8, i + +, Clear[b, u]; t[i] = 1 + i ∗ d; b[x− ] = u[i + 1][x];

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The Method of Lines and Duality Principles for Non-Convex Models

9. For[k = 1, k < 30, k + +, (we have fixed the number of iterations) 1 (b[x] − a) ∗ d z = u[i + 1][x] + b[x] + a + t[i] 2 + t[i]1 2 D[b[x], {x, 2}] ∗ d 2 + (−A ∗ b[x]3 + B ∗ u[x] + f [i]) ∗ de1 /3.0; z= Series[z, {u[i + 1][x], 0, 3}, {u[i + 1]′ [x], 0, 1}, {u[i + 1]′′ [x], 0, 1}, {u[i + 1]′′′ [x], 0, 0}, {u[i + 1]′′′′ [x], 0, 0}]; z = Normal[z], z = Expand[z]; b[x− ] = z]; 10. a1 [i] = z; 11. Clear[b]; 12. u[i + 1][x− ] = b[x]; 13. a = a1 [i] ]; 14. b[x− ] = u f [x]; 15. For[i = 1, i < m8, i + +, A1 = a1 [m8 − i]; A1 = Series[A1 , {u f [x], 0, 3}, {u′f [x], 0, 1}, {u′′f [x], 0, 1}, {u′′′f [x], 0, 0}, {u′′′′ f [x], 0, 0}]; A1 = Normal[A1 ]; A1 = Expand[A1 ]; u[m8 − i][x− ] = A1 ; b[x− ] = A1 ]; Print[u[m8/2][x]]; ************************************* The numerical expressions for the solutions of the concerning N = 10 lines are given by u[1][x] = 0.47352 + 0.00691u f [x] − 0.00459u f [x]2 + 0.00265u f [x]3 + 0.00039(u′′f )[x] −0.00058u f [x](u′′f )[x] + 0.00050u f [x]2 (u′′f )[x] − 0.000181213u f [x]3 (u′′f )[x] (12.50) u[2][x] = 0.76763 + 0.01301u f [x] − 0.00863u f [x]2 + 0.00497u f [x]3 + 0.00068(u′′f )[x] −0.00103u f [x](u′′f )[x] + 0.00088u f [x]2 (u′′f )[x] − 0.00034u f [x]3 (u′′f )[x]

(12.51)

Dual Variational Formulation for a Large Class of Non-Convex Models

u[3][x]

=

■

283

0.91329 + 0.02034u f [x] − 0.01342u f [x]2 + 0.00768u f [x]3 + 0.00095(u′′f )[x] −0.00144u f [x](u′′f )[x] + 0.00122u f [x]2 (u′′f )[x] − 0.00051u f [x]3 (u′′f )[x] (12.52)

u[4][x]

0.97125 + 0.03623u f [x] − 0.02328u f [x]2 + 0.01289u f [x]3 + 0.00147331(u′′f )[x]

=

−0.00223u f [x](u′′f )[x] + 0.00182u f [x]2 (u′′f )[x] − 0.00074u f [x]3 (u′′f )[x] (12.53)

u[5][x]

=

1.01736 + 0.09242u f [x] − 0.05110u f [x]2 + 0.02387u f [x]3 + 0.00211(u′′f )[x] −0.00378u f [x](u′′f )[x] + 0.00292u f [x]2 (u′′f )[x] − 0.00132u f [x]3 (u′′f )[x] (12.54)

u[6][x]

=

1.02549 + 0.21039u f [x] − 0.09374u f [x]2 + 0.03422u f [x]3 + 0.00147(u′′f )[x] −0.00634u f [x](u′′f )[x] + 0.00467u f [x]2 (u′′f )[x] − 0.00200u f [x]3 (u′′f )[x] (12.55)

u[7][x] = 0.93854 + 0.36459u f [x] − 0.14232u f [x]2 + 0.04058u f [x]3 + 0.00259(u′′f )[x] −0.00747373u f [x](u′′f )[x] + 0.0047969u f [x]2 (u′′f )[x] − 0.00194u f [x]3 (u′′f )[x] (12.56)

u[8][x]

=

0.74649 + 0.57201u f [x] − 0.17293u f [x]2 + 0.02791u f [x]3 + 0.00353(u′′f )[x] −0.00658u f [x](u′′f )[x] + 0.00407u f [x]2 (u′′f )[x] − 0.00172u f [x]3 (u′′f )[x] (12.57)

u[9][x]

=

0.43257 + 0.81004u f [x] − 0.13080u f [x]2 + 0.00042u f [x]3 + 0.00294(u′′f )[x] −0.00398u f [x](u′′f )[x] + 0.00222u f [x]2 (u′′f )[x] − 0.00066u f [x]3 (u′′f )[x] (12.58)

12.7.3

Some plots concerning the numerical results

In this section, we present the lines 2, 4, 6, 8 related to results obtained in the last section. Indeed, we present such mentioned lines, in a first step, for the previous results obtained through the generalized method of lines and, in a second step, through a numerical method which is the combination of the Newton’s one and the generalized method of lines. In a third step, we also present the graphs by considering the expression of the lines as those also obtained through the generalized method of lines, up to the numerical coefficients for each function term, which are obtained by the numerical optimization of the functional J, below specified. We consider the case in which u0 (x) = 0 and u f (x) = sin(x).

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For the procedure mentioned above as the third step, recalling that N = 10 lines, considering that u′′f (x) = −u f (x), we may approximately assume the following general line expressions: un (x) = a(1, n) + a(2, n)u f (x) + a(3, n)u f (x)3 + a(4, n)u f (x)3 , ∀n ∈ {1, · · · N − 1}. Defining Wn = −e1

(un+1 (x) − 2un (x) + un−1 (x)) 2 e1 (un (x) − un−1 (x)) e1 ′′ − − 2 un (x) + un (x)3 − un (x) − 1, d tn d tn

and

N−1 Z 2π

J({a( j, n)}) =

∑

n=1 0

(Wn )2 dx

we obtain {a( j, n)} by numerically minimizing J. Hence, we have obtained the following lines for these cases. For such graphs, we have considered 300 nodes in x, with 2π/300 as units in x ∈ [0, 2π]. For the Lines 2, 4, 6, 8, through the generalized method of lines, please see Figures 12.1, 12.4, 12.7, 12.10. For the Lines 2, 4, 6, 8, through a combination of the Newton’s and the generalized method of lines, please see Figures 12.2, 12.5, 12.8, 12.11. Finally, for the Line 2, 4, 6, 8 obtained through the minimization of the functional J, please see Figures 12.3, 12.6, 12.9, 12.12.

0.78 0.775 0.77 0.765 0.76 0.755 0.75 0.745 0.74 0

50

100

150

200

250

300

Figure 12.1: Line 2, solution u2 (x) through the general method of lines.

Dual Variational Formulation for a Large Class of Non-Convex Models

0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.8 0

50

100

150

200

250

300

Figure 12.2: Line 2, solution u2 (x) through the Newton’s method.

0.9 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.8 0

50

100

150

200

250

300

Figure 12.3: Line 2, solution u2 (x) through the minimization of functional J.

■

285

286

■

The Method of Lines and Duality Principles for Non-Convex Models

1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9 0

50

100

150

200

250

300

Figure 12.4: Line 4, solution u4 (x) through the general method of lines.

1.2 1.18 1.16 1.14 1.12 1.1 1.08 1.06 1.04 1.02 1 0

50

100

150

200

250

300

Figure 12.5: Line 4, solution u4 (x) through the Newton’s method.

Dual Variational Formulation for a Large Class of Non-Convex Models

1.25

1.2

1.15

1.1

1.05

1

0.95 0

50

100

150

200

250

300

Figure 12.6: Line 4, solution u4 (x) through the minimization of functional J.

1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0

50

100

150

200

250

300

Figure 12.7: Line 6, solution u6 (x) through the general method of lines.

■

287

288

■

The Method of Lines and Duality Principles for Non-Convex Models

1.3

1.2

1.1

1

0.9

0.8

0.7 0

50

100

150

200

250

300

Figure 12.8: Line 6, solution u6 (x) through the Newton’s method.

1.3

1.2

1.1

1

0.9

0.8

0.7 0

50

100

150

200

250

300

Figure 12.9: Line 6, solution u6 (x) through the minimization of functional J.

Dual Variational Formulation for a Large Class of Non-Convex Models

1.2

1

0.8

0.6

0.4

0.2

0

-0.2 0

50

100

150

200

250

300

Figure 12.10: Line 8, solution u8 (x) through the general method of lines.

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

300

Figure 12.11: Line 8, solution u8 (x) through the Newton’s method.

■

289

290

■

The Method of Lines and Duality Principles for Non-Convex Models

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

50

100

150

200

250

300

Figure 12.12: Line 8, solution u8 (x) through the minimization of functional J.

12.8

Conclusion

In the first part of this article, we develop duality principles for non-convex variational optimization. In the final concerning sections, we propose dual convex formulations suitable for a large class of models in physics and engineering. In the last article section, we present an advance concerning the computation of a solution for a partial differential equation through the generalized method of lines. In particular, in its previous versions, we used to truncate the series in d 2 however, we have realized the results are much better by taking line solutions in series for u f [x] and its derivatives, as it is indicated in the present software. This is a little difference concerning the previous procedure, but with a great result improvement as the parameter ε > 0 is small. Indeed, with a sufficiently large N (number of lines), we may obtain very good qualitative results even as ε > 0 is very small.

Chapter 13

A Note on the Korn’s Inequality in a N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model

13.1

Introduction

In this article, we present a proof for the Korn inequality in Rn . The results are based on the standard tools of functional analysis and the Sobolev spaces theory. We emphasize such a proof is relatively simple and easy to follow since it is established in a very transparent and clear fashion. Remark 13.1.1 This chapter has been accepted for publication in a similar article format by the MDPI Journal Applied Mathematics, reference [21]: F.S. Botelho, A Note on Korn’s Inequality in an N-Dimensional Context and a Global Existence Result for a Non-Linear Plate Model. Applied Math 2023, 1: 1–11. About the references, we highlight, related results in a three dimensional context may be found in [54]. Other important classical results on the Korn’s inequality and concerning applications to models in elasticity may be found in [30, 31, 32].

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Remark 13.1.2 Generically throughout the text we denote Z 1/2 ∥u∥0,2,Ω = , ∀u ∈ L2 (Ω), |u|2 dx Ω

and

!1/2

n

∥u∥0,2,Ω =

∑

∥u j ∥20,2,Ω

, ∀u = (u1 , . . . , un ) ∈ L2 (Ω; Rn ).

j=1

Moreover, !1/2

n

∥u∥1,2,Ω =

∥u∥20,2,Ω + ∑ ∥ux j ∥20,2,Ω

, ∀u ∈ W 1,2 (Ω),

j=1

where we shall also refer throughout the text to the well known corresponding analogous norm for u ∈ W 1,2 (Ω; Rn ). At this point, we first introduce the following definition. Definition 13.1.3 Let Ω ⊂ Rn be an open, bounded set. We say that ∂ Ω is Cˆ 1 if such a manifold is oriented and for each x0 ∈ ∂ Ω, denoting xˆ = (x1 , ..., xn−1 ) for a local coordinate system compatible with the manifold ∂ Ω orientation, there exist r > 0 and a function f (x1 , ..., xn−1 ) = f (x) ˆ such that W = Ω ∩ Br (x0 ) = {x ∈ Br (x0 ) | xn ≤ f (x1 , ..., xn−1 )}. Moreover f (x) ˆ is a Lipschitz continuous function, so that | f (x) ˆ − f (y)| ˆ ≤ C1 |xˆ − y| ˆ 2 , on its domain, for some C1 > 0. Finally, we assume

∂ f (x) ˆ ∂ xk

n−1 k=1

is classically defined, almost everywhere also on its concerning domain, so that f ∈ W 1,2 . Remark 13.1.4 This mentioned set Ω is of a Lipschitzian type, so that we may refer to such a kind of sets as domains with a Lipschitzian boundary, or simply as Lipschitzian sets. At this point, we recall the following result found in [13], at page 222 in its Chapter 11. Theorem 13.1.5 Assume Ω ⊂ Rn is an open bounded set, and that ∂ Ω is Cˆ 1 . Let 1 ≤ p < ∞, and let V be a bounded open set such that Ω ⊂⊂ V . Then there exists a bounded linear operator E : W 1,p (Ω) → W 1,p (Rn ),

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such that for each u ∈ W 1,p (Ω) we have: 1. Eu = u, a.e. in Ω, 2. Eu has support in V , and 3. ∥Eu∥1,p,Rn ≤ C∥u∥1,p,Ω , where the constant depend only on p, Ω, and V. Remark 13.1.6 Considering the proof of such a result, the constant C > 0 may be also such that ∥ei j (Eu)∥0,2,V ≤ C(∥ei j (u)∥0,2,Ω + ∥u∥0,2,Ω ), ∀u ∈ W 1,2 (Ω; Rn ), ∀i, j ∈ {1, . . . , n}, for the operator e : W 1,2 (Ω; Rn ) → L2 (Ω; Rn×n ) specified in the next theorem. Finally, as the meaning is clear, we may simply denote Eu = u.

13.2

The main results, the Korn inequalities

Our main result is summarized by the following theorem. Theorem 13.2.1 Let Ω ⊂ Rn be an open, bounded and connected set with a Cˆ 1 (Lipschitzian) boundary ∂ Ω. Define e : W 1,2 (Ω; Rn ) → L2 (Ω; Rn×n ) by e(u) = {ei j (u)} where

1 ei j (u) = (ui, j + u j,i ), ∀i, j ∈ {1, . . . , n}, 2 and where generically, we denote ui, j =

∂ ui , ∀i, j ∈ {1, · · · , n}. ∂xj

Define also, n

∥e(u)∥0,2,Ω =

!1/2

n

∑∑

∥ei j(u) ∥20,2,Ω

.

i=1 j=1

Let L ∈ R+ be such V = [−L, L]n is also such that Ω ⊂ V 0 . Under such hypotheses, there exists C(Ω, L) ∈ R+ such that ∥u∥1,2,Ω ≤ C(Ω, L) (∥u∥0,2,Ω + ∥e(u)∥0,2,Ω ) , ∀u ∈ W 1,2 (Ω; Rn ).

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Proof 13.1 Suppose, to obtain contradiction, the concerning claim does not hold. Hence, for each k ∈ N there exists uk ∈ W 1,2 (Ω; Rn ) such that ∥uk ∥1,2,Ω > k (∥uk ∥0,2,Ω + ∥e(uk )∥0,2,Ω ) . In particular defining vk =

uk ∥uk ∥1,2,Ω

we obtain ∥vk ∥1,2,Ω = 1 > k (∥vk ∥0,2,Ω + ∥e(vk )∥0,2,Ω ) , so that

1 (∥vk ∥0,2,Ω + ∥e(vk )∥0,2,Ω ) < , ∀k ∈ N. k From this we have got, 1 ∥vk ∥0,2,Ω < , k and 1 ∥ei j (vk )∥0,2,Ω < , ∀k ∈ N, k so that ∥vk ∥0,2,Ω → 0, as k → ∞, and ∥ei j (vk )∥0,2,Ω → 0, as k → ∞. In particular ∥(vk ) j, j ∥0,2,Ω → 0, ∀ j ∈ {1, . . . , n}. At this point we recall the following identity in the distributional sense, found in [31], page 12, ∂ j (∂l vi ) = ∂ j eil (v) + ∂l ei j (v) − ∂i e jl (v), ∀i, j, l ∈ {1, . . . , n}. Fix j ∈ {1, . . . , n} and observe that ∥(vk ) j ∥1,2,V ≤ C∥(vk ) j ∥1,2,Ω , so that

1 C ≥ , ∀k ∈ N. ∥(vk ) j ∥1,2,V ∥(vk ) j ∥1,2,Ω

(13.1)

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Hence, ∥(vk ) j ∥1,2,Ω n o = sup ⟨∇(vk ) j , ∇ϕ⟩L2 (Ω) + ⟨(vk ) j , ϕ⟩L2 (Ω) : ∥ϕ∥1,2,Ω ≤ 1 ϕ∈C1 (Ω)

(vk ) j ∥(vk ) j ∥1,2,Ω L2 (Ω) (vk ) j + (vk ) j , ∥(vk ) j ∥1,2,Ω L2 (Ω) ! (vk ) j (vk ) j + (vk ) j , ≤ C ∇(vk ) j , ∇ ∥(vk ) j ∥1,2,V ∥(vk ) j ∥1,2,V L2 (V ) L2 (V ) n o = C sup ⟨∇(vk ) j , ∇ϕ⟩L2 (V ) + ⟨(vk ) j , ϕ⟩L2 (V ) : ∥ϕ∥1,2,V ≤ 1 . (13.2)

=

∇(vk ) j , ∇

ϕ∈Cc1 (V )

Here, we recall that C > 0 is the constant concerning the extension Theorem 13.1.5. From such results and (13.1), we have that n o sup ⟨∇(vk ) j , ∇ϕ⟩L2 (Ω) + ⟨(vk ) j , ϕ⟩L2 (Ω) : ∥ϕ∥1,2,Ω ≤ 1 ϕ∈C1 (Ω)

≤ C sup ϕ∈Cc1 (V )

= C sup ϕ∈Cc1 (V )

n o ⟨∇(vk ) j , ∇ϕ⟩L2 (V ) + ⟨(vk ) j , ϕ⟩L2 (V ) : ∥ϕ∥1,2,V ≤ 1 n ⟨e jl (vk ), ϕ,l ⟩L2 (V ) + ⟨e jl (vk ), ϕ,l ⟩L2 (V )

o −⟨ell (vk ), ϕ, j ⟩L2 (V ) + ⟨(vk ) j , ϕ⟩L2 (V ) , : ∥ϕ∥1,2,V ≤ 1 .

(13.3)

Therefore ∥(vk ) j ∥(W 1,2 (Ω)) sup {⟨∇(vk ) j , ∇ϕ⟩L2 (Ω) + ⟨(vk ) j , ϕ⟩L2 (Ω) : ∥ϕ∥1,2,Ω ≤ 1}

=

ϕ∈C1 (Ω)

!

n

≤ C

∑ ∥e jl (vk )∥0,2,V + ∥ell (vk )∥0,2,V + ∥(vk ) j ∥0,2,V l=1

!

n

≤ C1

∑ ∥e jl (vk )∥0,2,Ω + ∥ell (vk )∥0,2,Ω + ∥(vk ) j ∥0,2,Ω l=1

0 and C2 > 0. Summarizing, ∥(vk ) j ∥(W 1,2 (Ω))

0. Assume also Γ0 is such that for each j ∈ {1, · · · , n} and each x = (x1 , · · · , xn ) ∈ Ω there exists x0 = ((x0 )1 , · · · , (x0 )n ) ∈ Γ0 such that (x0 )l = xl , ∀l ̸= j, l ∈ {1, · · · , n}, and the line Ax0 ,x ⊂ Ω where Ax0 ,x = {(x1 , · · · , (1 − t)(x0 ) j + tx j , · · · , xn ) : t ∈ [0, 1]}. Under such hypotheses, there exists C(Ω, L) ∈ R+ such that ∥u∥1,2,Ω ≤ C(Ω, L) ∥e(u)∥0,2,Ω , ∀u ∈ Hˆ 0 . Proof 13.2 Suppose, to obtain contradiction, the concerning claim does not hold. Hence, for each k ∈ N there exists uk ∈ Hˆ 0 such that ∥uk ∥1,2,Ω > k ∥e(uk )∥0,2,Ω . In particular defining vk =

uk ∥uk ∥1,2,Ω

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similarly to the proof of the last theorem, we may obtain ∥(vk ) j, j ∥0,2,Ω → 0, as k → ∞, ∀ j ∈ {1, . . . , n}. From this, the hypotheses on Γ0 and from the standard Poincar´e inequality proof we obtain ∥(vk ) j ∥0,2,Ω → 0, as k → ∞, ∀ j ∈ {1, . . . , n}. Thus, also similarly as in the proof of the last theorem, we may infer that ∥vk ∥1,2,Ω → 0, as k → ∞, which contradicts ∥vk ∥1,2,Ω = 1, ∀k ∈ N. The proof is complete.

13.3

An existence result for a non-linear model of plates

In the present section, as an application of the results on the Korn’s inequalities presented in the previous sections, we develop a new global existence proof for a Kirchhoff-Love thin plate model. Previous results on existence in mathematical elasticity and related models may be found in [30, 31, 32]. At this point we start to describe the primal formulation. Let Ω ⊂ R2 be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of Ω, which is assumed to be regular (Lipschitzian), is denoted by ∂ Ω. The vectorial basis related to the cartesian system {x1 , x2 , x3 } is denoted by (aα , a3 ), where α = 1, 2 (in general Greek indices stand for 1 or 2), and where a3 is the vector normal to Ω, whereas a1 and a2 are orthogonal vectors parallel to Ω. Also, n is the outward normal to the plate surface. The displacements will be denoted by uˆ = {uˆα , uˆ3 } = uˆα aα + uˆ3 a3 . The Kirchhoff-Love relations are uˆα (x1 , x2 , x3 ) = uα (x1 , x2 ) − x3 w(x1 , x2 ),α and uˆ3 (x1 , x2 , x3 ) = w(x1 , x2 ).

(13.5)

Here −h/2 ≤ x3 ≤ h/2 so that we have u = (uα , w) ∈ U where U = (uα , w) ∈ W 1,2 (Ω; R2 ) ×W 2,2 (Ω), ∂w = 0 on ∂ Ω} uα = w = ∂n = W01,2 (Ω; R2 ) ×W02,2 (Ω). It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

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We define the operator Λ : U → Y ×Y , where Y = Y ∗ = L2 (Ω; R2×2 ), by Λ(u) = {γ(u), κ(u)}, uα,β + uβ ,α w,α w,β + , 2 2 καβ (u) = −w,αβ .

γαβ (u) =

The constitutive relations are given by Nαβ (u) = Hαβ λ µ γλ µ (u),

(13.6)

Mαβ (u) = hαβ λ µ κλ µ (u),

(13.7)

h2

where: {Hαβ λ µ } and {hαβ λ µ = 12 Hαβ λ µ }, are symmetric positive definite fourth order tensors. From now on, we denote {H αβ λ µ } = {Hαβ λ µ }−1 and {hαβ λ µ } = {hαβ λ µ }−1 . Furthermore, {Nαβ } denote the membrane force tensor and {Mαβ } the moment one. The plate stored energy, represented by (G ◦ Λ) : U → R, is expressed by (G ◦ Λ)(u) =

1 2

Z

Nαβ (u)γαβ (u) dx + Ω

1 2

Z

Mαβ (u)καβ (u) dx

(13.8)

Ω

and the external work, represented by F : U → R, is given by F(u) = ⟨w, P⟩L2 (Ω) + ⟨uα , Pα ⟩L2 (Ω) ,

(13.9)

where P, P1 , P2 ∈ L2 (Ω) are external loads in the directions a3 , a1 and a2 respectively. The potential energy, denoted by J : U → R, is expressed by: J(u) = (G ◦ Λ)(u) − F(u) Finally, we also emphasize that from now on, as their meaning are clear, we may denote L2 (Ω) and L2 (Ω; R2×2 ) simply by L2 , and the respective norms by ∥ · ∥2 . Moreover derivatives are always understood in the distributional sense, 0 may denote the zero vector in appropriate Banach spaces and, the following and relating notations are used: ∂w , w,α = ∂ xα w,αβ =

∂ 2w , ∂ xα ∂ xβ

uα,β =

∂ uα , ∂ xβ

Nαβ ,1 =

∂ Nαβ , ∂ x1

Nαβ ,2 =

∂ Nαβ . ∂ x2

and

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On the existence of a global minimizer

At this point we present an existence result concerning the Kirchhoff-Love plate model. We start with the following two remarks. Remark 13.1 Let {Pα } ∈ L∞ (Ω; R2 ). We may easily obtain by appropriate Lebesgue integration {T˜αβ } symmetric and such that T˜αβ ,β = −Pα , in Ω. Indeed, extending {Pα } to zero outside Ω if necessary, we may set T˜11 (x, y) = − T˜22 (x, y) = −

Z x 0

P1 (ξ , y) dξ ,

Z y 0

P2 (x, ξ ) dξ ,

and T˜12 (x, y) = T˜21 (x, y) = 0, in Ω. Thus, we may choose a C > 0 sufficiently big, such that {Tαβ } = {T˜αβ +Cδαβ } is positive definite in Ω, so that Tαβ ,β = T˜αβ ,β = −Pα , where {δαβ } is the Kronecker delta. So, for the kind of boundary conditions of the next theorem, we do NOT have any restriction for the {Pα } norm. Summarizing, the next result is new and it is really a step forward concerning the previous one in Ciarlet [31]. We emphasize, this result and its proof through such a tensor {Tαβ } are new, even though the final part of the proof is established through a standard procedure in the calculus of variations. Finally, more details on the Sobolev spaces involved may be found in [1, 12, 13, 35]. Related duality principles are addressed in [34, 12, 13]. At this point we present the main theorem in this section. Theorem 13.4.1 Let Ω ⊂ R2 be an open, bounded, connected set with a Lipschitzian boundary denoted by ∂ Ω = Γ. Suppose (G ◦ Λ) : U → R is defined by G(Λu) = G1 (γ(u)) + G2 (κ(u)), ∀u ∈ U,

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where G1 (γu) = and G2 (κu) =

1 2

Z

1 2

Z

Hαβ λ µ γαβ (u)γλ µ (u) dx, Ω

hαβ λ µ καβ (u)κλ µ (u) dx, Ω

where Λ(u) = (γ(u), κ(u)) = ({γαβ (u)}, {καβ (u)}), uα,β + uβ ,α w,α w,β + , 2 2 καβ (u) = −w,αβ ,

γαβ (u) =

and where J(u) = W (γ(u), κ(u)) − ⟨Pα , uα ⟩L2 (Ω) −⟨w, P⟩L2 (Ω) − ⟨Pαt , uα ⟩L2 (Γt ) −⟨Pt , w⟩L2 (Γt ) ,

(13.10)

where, U

= {u = (uα , w) = (u1 , u2 , w) ∈ W 1,2 (Ω; R2 ) ×W 2,2 (Ω) : ∂w = 0, on Γ0 }, uα = w = ∂n

(13.11)

where ∂ Ω = Γ0 ∪ Γt and the Lebesgue measures mΓ (Γ0 ∩ Γt ) = 0, and mΓ (Γ0 ) > 0. We also define, F1 (u) =

−⟨w, P⟩L2 (Ω) − ⟨uα , Pα ⟩L2 (Ω) − ⟨Pαt , uα ⟩L2 (Γt ) −⟨Pt , w⟩L2 (Γt ) + ⟨εα , u2α ⟩L2 (Γt )

≡ −⟨u, f⟩L2 + ⟨εα , u2α ⟩L2 (Γt ) ≡ −⟨u, f1 ⟩L2 − ⟨uα , Pα ⟩L2 (Ω) + ⟨εα , u2α ⟩L2 (Ω) , where ⟨u, f1 ⟩L2 = ⟨u, f⟩L2 − ⟨uα , Pα ⟩L2 (Ω) , εα > 0, ∀α ∈ {1, 2} and f = (Pα , P) ∈ L∞ (Ω; R3 ).

(13.12)

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Let J : U → R be defined by J(u) = G(Λu) + F1 (u), ∀u ∈ U. Assume there exists {cαβ } ∈ R2×2 such that cαβ > 0, ∀α, β ∈ {1, 2} and G2 (κ(u)) ≥ cαβ ∥w,αβ ∥22 , ∀u ∈ U. Under such hypotheses, there exists u0 ∈ U such that J(u0 ) = min J(u). u∈U

Proof 13.3

Observe that we may find Tα = {(Tα )β } such that divTα = Tαβ ,β = −Pα ,

and also such that {Tαβ } is positive definite and symmetric (please, see Remark 13.1). Thus defining uα,β + uβ ,α 1 + w,α w,β , (13.13) vαβ (u) = 2 2 we obtain J(u)

=

G1 ({vαβ (u)}) + G2 (κ(u)) − ⟨u, f⟩L2 + ⟨εα , u2α ⟩L2 (Γt )

=

G1 ({vαβ (u)}) + G2 (κ(u)) + ⟨Tαβ ,β , uα ⟩L2 (Ω) − ⟨u, f1 ⟩L2 + ⟨εα , u2α ⟩L2 (Γt ) uα,β + uβ ,α G1 ({vαβ (u)}) + G2 (κ(u)) − Tαβ , 2 L2 (Ω)

=

=

+⟨Tαβ nβ , uα ⟩L2 (Γt ) − ⟨u, f1 ⟩L2 + ⟨εα , u2α ⟩L2 (Γt ) 1 − ⟨u, f1 ⟩L2 + ⟨εα , u2α ⟩L2 (Γt ) G1 ({vαβ (u)}) + G2 (κ(u)) − Tαβ , vαβ (u) − w,α w,β 2 L2 (Ω) +⟨Tαβ nβ , uα ⟩L2 (Γt )

≥

1

T , w,α w,β L2 (Ω) − ⟨u, f1 ⟩L2 + ⟨εα , u2α ⟩L2 (Γt ) + G1 ({vαβ (u)}) 2 αβ −⟨Tαβ , vαβ (u)⟩L2 (Ω) + ⟨Tαβ nβ , uα ⟩L2 (Γt ) . (13.14)

cαβ ∥w,αβ ∥22 +

From this, since {Tαβ } is positive definite, clearly J is bounded below. Let {un } ∈ U be a minimizing sequence for J. Thus there exists α1 ∈ R such that lim J(un ) = inf J(u) = α1 .

n→∞

u∈U

From (13.14), there exists K1 > 0 such that ∥(wn ),αβ ∥2 < K1 , ∀α, β ∈ {1, 2}, n ∈ N.

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Therefore, there exists w0 ∈ W 2,2 (Ω) such that, up to a subsequence not relabeled, (wn ),αβ ⇀ (w0 ),αβ , weakly in L2 , ∀α, β ∈ {1, 2}, as n → ∞. Moreover, also up to a subsequence not relabeled, (wn ),α → (w0 ),α , strongly in L2 and L4 ,

(13.15)

∀α, ∈ {1, 2}, as n → ∞. Also from (13.14), there exists K2 > 0 such that, ∥(vn )αβ (u)∥2 < K2 , ∀α, β ∈ {1, 2}, n ∈ N, and thus, from this, (13.13) and (13.15), we may infer that there exists K3 > 0 such that ∥(un )α,β + (un )β ,α ∥2 < K3 , ∀α, β ∈ {1, 2}, n ∈ N. From this and Korn’s inequality, there exists K4 > 0 such that ∥un ∥W 1,2 (Ω;R2 ) ≤ K4 , ∀n ∈ N. So, up to a subsequence not relabeled, there exists {(u0 )α } ∈ W 1,2 (Ω, R2 ), such that (un )α,β + (un )β ,α ⇀ (u0 )α,β + (u0 )β ,α , weakly in L2 , ∀α, β ∈ {1, 2}, as n → ∞, and, (un )α → (u0 )α , strongly in L2 , ∀α ∈ {1, 2}, as n → ∞. Moreover, the boundary conditions satisfied by the subsequences are also satisfied for w0 and u0 in a trace sense, so that u0 = ((u0 )α , w0 ) ∈ U. From this, up to a subsequence not relabeled, we get γαβ (un ) ⇀ γαβ (u0 ), weakly in L2 , ∀α, β ∈ {1, 2}, and καβ (un ) ⇀ καβ (u0 ), weakly in L2 , ∀α, β ∈ {1, 2}. Therefore, from the convexity of G1 in γ and G2 in κ we obtain inf J(u) =

u∈U

α1

= lim inf J(un ) n→∞

≥ J(u0 ). Thus, J(u0 ) = min J(u). u∈U

The proof is complete.

(13.16)

A Note on the Korn’s Inequality in a N-Dimensional Context

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Conclusion

In this chapter, we have developed a new proof for the Korn inequality in a specific ndimensional context. In the second text part, we present a global existence result for a non-linear model of plates. Both results represent some new advances concerning the present literature. In particular, the results for the Korn’s inequality so far known are for a three dimensional context such as in [54], for example, whereas we have here addressed a more general n-dimensional case. In a future research, we intend to address more general models, including the corresponding results for manifolds in Rn .

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Index A adjoint operator 138, 159 affine continuous 128, 151

B Banach space 74–77, 79–81, 85, 101, 103, 104, 109, 112, 126–132, 136, 138–142, 145, 157–159, 161, 167, 168, 172, 173, 177–179, 183, 186–188, 193, 200–202, 204, 206, 210, 220, 254, 255, 298 bi-polar functional 129 bounded function 27, 109, 147 bounded operator 159, 195, 231, 292 bounded set 292

C calculus of variations 10, 27, 73, 74, 98, 99, 109, 110, 124, 139, 155, 159, 200, 201, 210, 219, 220, 227 Cauchy sequence 75–78, 156, 177, 187 closed set 123, 202 compact set 78, 104, 142, 143, 152 compactness 151 cone 167, 168, 170, 171, 179, 183, 206 connected set 5, 27, 38, 39, 109, 123, 212, 220, 230, 233, 236, 238, 242, 245, 249, 254, 260, 263, 268, 273, 293, 296, 297, 299 constrained optimization 201

continuity 106, 111, 124, 126, 130, 139, 155, 204 continuous function 3, 27, 75, 123, 132, 143, 147, 292 convergence 4, 9, 21, 30, 44, 67, 69, 111 convex analysis 73, 126, 129, 151, 161, 212, 219, 222, 227, 253, 254 convex envelop 128, 129 convex function 83, 130 convex set 126, 169, 171, 182, 186, 202, 209, 238, 240, 244, 250

D dense set 148 dual space 201, 202, 220, 254 dual variational formulation 212, 216, 220, 224, 225, 232, 253, 254, 259, 260, 263, 268, 273 duality 27, 126, 136, 155, 161, 171, 211, 212, 216, 219, 220, 223, 225, 226, 253, 254, 256, 260, 263, 277, 290, 299 duality principle 27, 211, 212, 216, 219, 220, 223, 225, 253, 256, 260, 263, 290, 299 du Bois-Reymond lemma 96, 98, 99, 121

E Ekeland variational principle 155 elasticity 291, 297 epigraph 126, 128

312 ■ The Method of Lines and Duality Principles for Non-Convex Models

F finite dimensional space 213 fluid mechanics 36 function 2, 3, 11, 12, 13, 17, 18, 26, 27, 37, 67, 74, 75, 77, 79, 82–84, 87, 99, 103, 105, 106, 109, 114, 118, 120, 123, 126, 130, 132–134, 136, 138, 139, 143, 147–155, 172, 173, 177, 178, 188, 194, 195, 198, 201, 204, 234, 266, 278, 279, 280, 283, 292 functional 10, 27, 77–81, 83, 85, 86, 101, 103–105, 109, 110, 116, 123, 126–134, 139–142, 145, 155, 158, 159, 161, 168, 170–172, 193, 200–202, 205, 210, 212, 213, 219, 220, 223–225, 227, 232, 233, 236, 238, 243, 246, 249, 253–256, 259, 260, 264, 277, 283–285, 287, 288, 290, 291 functional analysis 10, 27, 200, 201, 210, 212, 219, 253, 277, 291

G Generalized Method of Lines 1–3, 7, 9–11, 17, 25–27, 36, 37, 48, 58, 65, 66, 186, 211, 253, 277, 278, 283, 284, 290 global existence 291, 297, 303 graph 81, 147, 283, 284

H Hahn Banach theorem 131, 182, 201, 202, 205, 210 Hahn Banach theorem, geometric form 182, 210 hyper-plane 131, 167, 169, 202

I inequality 74, 120, 170, 173, 175, 176, 178, 179, 181, 183, 186, 200, 206, 291, 293, 297, 302, 303 inner product 83 integration 110, 121, 124, 299

interior 127, 168, 169, 182, 209 inverse mapping theorem 173

L Lagrange multiplier 168, 170, 172, 178–180, 184, 185, 194, 200, 201, 204, 206, 207, 210, 214, 234 Lebesgue dominated convergence theorem 111 Lebesgue measurable set 296 Lebesgue space 202, 220, 254 Legendre functional 132, 133 Legendre Hadamard condition 112 Legendre transform 132–134, 217, 233, 258, 263, 271, 276 limit 78, 81, 102, 113, 120 limit point 81, 102 linear functional 201, 205 linear operator 134, 138, 159, 195, 206, 292 local minimum 103, 110, 117, 122, 172, 178, 179, 183–185, 204, 206 lower semi-continuous 126, 128, 142, 145, 146, 156–158, 261

M matrix version of generalized method of lines 7 maximal 129 maximum 227 measurable function 132 measurable set 149, 296 measure 27, 119, 147, 296, 300 metric 155, 157 metric space 155 minimizer 139, 155, 299 minimum 80, 91, 103, 110, 116, 117, 122, 142, 143, 172, 178, 179, 183–185, 204, 206, 220, 223, 226

N Navier-Stokes system 36–41, 48, 50, 54, 58, 60, 65 necessary conditions 103, 110, 116, 179, 183, 194, 262

Index ■ 313 neighborhood 126–128, 130, 133, 138, 147, 150, 151, 153, 154, 194, 198, 204, 217, 259 Newton’s method 3, 5–7, 9, 68, 69, 285, 286, 288, 289 norm 74–77, 128, 148, 177, 186, 187, 202, 220, 254, 292, 298, 299 normed space 74, 75 null space 173, 178 numerical 2–5, 9–11, 16, 26, 27, 30, 36–38, 41, 54, 58, 66, 69, 211, 219, 223–225, 229, 253, 277, 278, 282–284

O open set 132, 147–149, 152, 292 operator 40, 134, 138, 159, 186, 195, 206, 213, 231, 254, 292, 293, 298 optimality conditions 179, 200 optimization 27, 73, 136, 167, 168, 183, 184, 193, 200, 201, 204, 206, 210, 212, 219, 220, 223, 225, 226, 228, 253, 259, 283, 290

142, 143, 145–149, 151, 152, 163, 169, 171, 182, 186, 201, 202, 209, 212, 220, 224, 225, 230, 233, 236, 238, 240, 242, 244, 245, 249, 250, 254, 260, 263, 268, 273, 279, 280, 292, 293, 296, 297, 299 simple function 148 Sobolev space 11, 27, 38, 219, 254, 277, 291, 299 space 11, 27, 38, 74–77, 79–81, 85, 101, 103, 104, 109, 112, 126–132, 136, 138–142, 145, 149, 155, 157–159, 161, 167–169, 172, 173, 177–179, 183, 186–188, 193, 200–202, 204, 206, 210, 213, 219, 220, 254, 255, 277, 291, 298, 299 sufficient conditions 87, 92, 93, 103 symmetric operator 231, 298

T topological dual space 201, 220, 254 topological space 201, 220, 254 topology 126–128, 150, 157

P

U

phase transition 219, 220, 223, 225 plate model 230, 291, 297, 299 positive functional 168 positive operator 298 positive set 148, 231, 299, 301

unique 39, 40, 133, 142, 186, 191, 202, 220, 254 upper semi-continuous 142, 143, 145

R range 178, 206 real set 109 reflexive spaces 138, 142, 145, 159

weak topology 126 weakly closed 128 weakly compact 143–145 Weierstrass necessary condition 116 Weierstrass-Erdmann conditions 120

S

Z

set 5, 12, 13, 27, 38, 39, 44, 54, 57, 67, 71, 78, 101, 104–106, 109, 114, 119, 120, 123, 126, 127, 130, 132, 137,

Zero 9, 138, 147, 150, 151, 153, 169, 172, 298, 299

W