微分方程和微分几何(英文版)
作者: Louis Nirenberg
出版时间:2018-09-25
出版社:高等教育出版社
- 高等教育出版社
- 9787040503029
- 1版
- 227561
- 46253999-0
- 精装
- 16开
- 2018-09-25
- 300
- 376
- 理学
- 数学
- O186.1;O175
- 数学类
- 研究生及以上
Front Matter
Part I Existence Theorems in Partial Differential Equations
1 Preliminaries.
1.1 Introduction.
1.2 TheMaximumPrinciple
1.3 Consequences of theMaximumPrinciple
2 The Potential Equation
2.1 Fundamental Solution
2.2 The Poisson Integral Formula
2.3 TheMean Value Property of Potential Functions
2.4 Estimates of Derivatives of Harmonic Functions and Analyticity
2.5 The Theorems and Inequality ofHarnack
2.6 Theoremon Removable Singularities
3 The PerronMethod for Solving the Dirichlet Problem
3.1 The PerronMethod
3.2 The PerronMethod forMore General Elliptic Equations
4 SchauderEstimates.
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 BoundaryValue Problem
4.6 Strong Barrier Functions, and the Boundary Value Problem
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 Formof 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 Theorem2.1
3 On Surfaces with Constant Negative Gauss Curvature
3.1 Hilbert’s TheoremonHyperbolic 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 Theoremof Christoffel on Closed Surfaces
10 Minkowski’s Problem
11 Existence of a Closed Convex Surface Solving Minkowski’s Problem
About the author