A Variational Theory of Convolution-Type Functionals 9789819906840, 9789819906857
134 90 1MB
English Pages 122 Year 2023
Table of contents :
Preface
Contents
1 Introduction
References
2 Convolution-Type Energies
2.1 Notation
2.2 Setting of the Problem and Comments
2.3 Assumptions
Reference
3 The -Limit of a Class of Reference Energies
3.1 The -Limit of G[a]
References
4 Asymptotic Embedding and Compactness Results
4.1 An Extension Result
4.2 Control of Long-Range Interactions with Short-Range Interactions
4.3 Compactness in Lp Spaces
4.4 Poincaré Inequalities
References
5 A Compactness and Integral-Representation Result
5.1 The Integral-Representation Theorem
5.2 Truncated-Range Functionals
5.3 Fundamental Estimates
5.4 Proof of the Integral-Representation Theorem
5.5 Convergence of Minimum Problems
5.6 Euler-Lagrange Equations
5.6.1 Regularity of Functionals F
5.6.2 Relations with Minimum Problems
References
6 Periodic Homogenization
6.1 A Homogenization Theorem
6.2 The Convex Case
6.3 Relaxation of Convolution-Type Energies
6.4 An Extension Lemma from Periodic Lipschitz Domains
6.5 Homogenization on Perforated Domains
References
7 A Generalization and Applications to Point Clouds
7.1 Perturbed Convolution-Type Functionals
7.2 Application to Functionals Defined on Point Clouds
References
8 Stochastic Homogenization
References
9 Application to Convex Gradient Flows
9.1 The Minimizing-Movement Approach to Gradient Flows
9.2 Homogenized Flows for Convex Energies
References
Index
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