This book is a superbly written by a world leading expert on partial differential equations and differential geometry. It consists of two parts. Part I covers the existence and uniqueness of solutions of elliptic differential equations. It is direct to the point, everything moves smoothly and quickly, and there are no unnecessary discussions or digressions. Many topics discussed in Part II will be new and surprising to many students, even to some experts in differential geometry. Both the selection of topics and the exposition are excellent. The detailed discussion of the case of surfaces motivated the later analogues in the higher dimensions. 本书是著名的数学家在其研究鼎盛时期所写的有关偏微分方程和微分几何的著作。分成两部分。第I部分讨论了椭圆微分方程解的存在性和唯一性，简洁、切中要点，叙述流畅，体现了著名数学家对所写材料的深刻理解。第II部分所讨论的议题对于很多读者，甚至微分几何方面的专家都是新的。 本书篇幅短小精悍，却包含丰富的内容。可供对本领域感兴趣的读者参考。

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