Fully Nonlinear Elliptic Equations (Colloquium Publications (Amer Mathematical Soc))

This book provides a self-contained development of the regularity theory for solutions of fully nonlinear elliptic equations. Caffarelli and Cabré offer a detailed presentation of all techniques needed to extend the classical Schauder and Calderón-Zygmund regularity theories for linear elliptic equations to the fully nonlinear context. The authors present the key ideas and prove all the results needed for the regularity theory of viscosity solutions of fully nonlinear equations. The book contains the study of convex fully nonlinear equations and fully nonlinear equations with variable coefficients. The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov–Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients. Introduction Chapter 1. Preliminaries Chapter 2. Viscosity Solutions Of Elliptic Equations Chapter 3. Alexandroff Estimate And Maximum Principle Chapter 4. Harnack Inequality Chapter 5. Uniqueness Of Solutions Chapter 6. Concave Equations Chapter 7. $w^{2,p}$ Regularity Chapter 8. Hölder Regularity Chapter 9. The Dirichlet Problem For Concave Equations Luis A. Caffarelli, Xavier Cabré. Includes Bibliographical References (p. 99-101). Cover Title page Contents Introduction Chapter 1. Preliminaries Chapter 2. Viscosity solutions of elliptic equations Chapter 3. Alexandroff estimate and maximum principle Chapter 4. Harnack inequality Chapter 5. Uniqueness of solutions Chapter 6. Concave equations Chapter 7. W^{2,p} Regularity Chapter 8. Hölder regularity Chapter 9. The Dirichlet problem for concave equations Bibliography Index Back Cover Presents a self-contained development of the regularity theory for solutions of fully nonlinear elliptic equations. This book describes various techniques that are needed to extend the classical Schauder and Calderon-Zygmund regularity theories for linear elliptic equations to the fully nonlinear context.