Lectures on Differential Equations and Differential Geometry 9787040503029, 7040503026
394 79 1MB
English Pages 178 Year 2018
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
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
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