This revision of the 1983 second edition of"Elliptic Partial Differential Equations of Second Order" corresponds to the Russian edition, published in 1989, in which we essentially updated the previous version to 1984. The additional text relates to the boundary H61der derivative estimates of Nikolai Krylov, which provided a fundamental component of the further development of the classical theory of elliptic (and parabolic), fully nonlinear equations in higher dimensions. In our presentation we adapted a simplification of Krylov's approach due to Luis Caffarelli.

Chapter 1 Introduction

Part Ⅰ Linear Equations

Chapter 2 Laplace’s Equation

Chapter 3 The Classical Maximum Principle

Chapter 4 Poisson's Equation and the Newtonian Potential

Chapter 5 Banach and Hilbert Spaces

Chapter 6 Calssical Solutions; the Schauder Approach

Chapter 7 Sobolev Spaces

Chapter 8 Generalized Solutiona and regularity

Chapter 9 Strong Solutions

Part Ⅱ Quasilinear Equations

Chapter 10 Maximum and Comparison Principles

Chapter 11 Topological Fixed Point Theorems and Their Application

Chapter 12 Equation in Two Varables

Chapter 13 Holder Extimates for the Cradient

Chapter 14 Boundary Gradient Estimates

Chapter 15 Global and Interior Gradient Bounds

Chapter 16 Equations of Mean Curvature Type

Chapter 17 Fully Nonlinear Equations

Bibliography

Epilogue

Subject Index

Notation Index