Lectures on Differential Equations and Differential Geometry 9787040503029, 7040503026

Table of contents :
Preface
Contents
Part I. Existence Theorems in Partial Differential Equations
1. Preliminaries
1.1 Introduction
1.2 The Maximum Principle
1.3 Consequences of the Maximum Principle
2. The Potential Equation
2.1 Fundamental Solution
2.2 The Poisson Integral Formula
2.3 The Mean Value Property of Potential Functions
2.4 Estimates of Derivatives of Harmonic Functions and Analyticity
2.5 The Theorems and Inequality of Harnack
2.6 Theorem on Removable Singularities
3. The Perron Method for Solving the Dirichlet Problem
3.1 The Perron Method
3.2 The Perron Method for More General Elliptic Equations
4. Schauder Estimates
4.1 Poisson's Equation
4.2 A Preliminary Estimate
4.3 Statement of Schauder's Estimates
4.4 Some Applications of the Interior Estimates
4.5 The Boundary Value Problem
4.6 Strong Barrier Functions, and the Boundary Value Problem in Non-smooth Domains
5. Derivation of the Schauder Estimates
5.1 A Preliminary Estimate
5.2 A Further Investigation of the Poisson Equation
5.3 Completion of the Interior Estimates
Part II. Seminar on Differential Geometry in the Large
1. Complete Surfaces
2. The Form of Complete Surfaces of Positive Gauss Curvature in Three-dimensional Space
2.1 Hadamard's Principle
2.2 Completeness of a Surface
2.3 Examples Showing that the Properties V, V' and E are Independent
2.4 Main Theorem
2.5 Consequence
2.6 Analogous Theorems for Plane Curves
2.7 Proof of Theorem 2.1
3. On Surfaces with Constant Negative Gauss Curvature
3.1 Hilbert's Theorem on Hyperbolic Surfaces
3.2 Asymptotic Coordinates in the Small
3.3 Considerations in the Large
3.4 Bounds on the Extended Angle Function
4. Isometric Deformations in the Small
5. Rigidity of Closed Convex Surfaces
6. Rigid Open Convex Surfaces
7. Rigidity of Sphere
8. Uniqueness of Closed Convex Surfaces with Prescribed Line Element
9. A Theorem of Christoffel on Closed Surfaces
10. Minkowski's Problem
11. Existence of a Closed Convex Surface Solving Minkowski's Problem
About the author

Polecaj historie

Lectures on Differential Equations and Differential Geometry (Classical Topics in Mathematics) [1 ed.] 7040503026, 9787040503029

This book is superbly written by a world-leading expert on partial differential equations and differential geometry. It

701 130 1MB Read more

Lectures on elliptic partial differential equations

779 76 558KB Read more

Lectures on Differential Equations [1 ed.]

257 81 17MB Read more

Lectures on differential equations 9781470451738, 1470451735

2,756 151 3MB Read more

Lectures on Differential Geometry 1571460128, 9781571460127

In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced Studies in Princet

879 62 83MB Read more

Lectures on nonsmooth differential geometry 9783030386122, 9783030386139

612 91 1MB Read more

Lectures on Differential Geometry 1571460128, 9781571460127

In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced Studies in Princet

437 57 33MB Read more

Differential Geometry, Differential Equations, and Mathematical Physics 9783030632526, 9783030632533

1,455 324 3MB Read more

Differential Geometry, Differential Equations, and Mathematical Physics 9783030632533

716 125 15MB Read more

Lectures on Differential Equations. (AM-14), Volume 14 9781400881949

The description for this book, Lectures on Differential Equations. (AM-14), Volume 14, will be forthcoming.

195 55 9MB Read more