Fully nonlinear elliptic equations. (English) Zbl 0834.35002

The present monograph is devoted to the study of fully nonlinear elliptic equations of the form \[ F(D^2 u,x)= f(x), \qquad x\in \Omega \subset \mathbb{R}^n,\tag{1} \] where \(u: \Omega\to \mathbb{R}\), \(D^2 u= (\partial^2 u/ \partial x_i \partial x_j )_{ij}\), \(F\) and \(f\) are continuous in \(x\). Let \(S\) denote the set of symmetric \(n\times n\) matrices. \(F\) is assumed to satisfy the uniform ellipticity condition \[ \lambda|N|\leq F(M+ N,x)- F(M, x)\leq \Lambda|N| \] for every \(x\in \Omega\), \(M\in S\) and every not negative definite \(N\in S\). Examples of such equations are Pucci’s equation, the Bellman equation, Isaacs equation and the Monge-Ampère equation. The authors are mainly concerned with viscosity (sub-, super-) solutions of (1). The fundamental definitions and notations are given in Chapters 1 and 2. In Chapter 3 the Alexandrov-Bakelman-Pucci estimate is proved for viscosity supersolutions. This estimate is then used in Chapter 4 to prove a weak Harnack inequality for supersolutions, a local maximum principle for subsolutions and the strong maximum principle for supersolutions. In addition, interior Hölder regularity and Hölder continuity up to the boundary (for continuous boundary data) are obtained. In Chapter 5 uniqueness and \(C^{1, \alpha}\) regularity are studied for the Dirichlet problem \[ F(D^2 u)=0 \quad \text{in } \Omega, \qquad u=g \quad \text{on }\partial \Omega. \tag{2} \] For concave functions \(F\), interior \(C^{2, \alpha}\) regularity for viscosity solutions of (2) is proved in Chapter 6. For solutions of (1), \(W^{2,p}\) regularity and a priori estimates \((p> n)\) are proved in Chapter 7 and interior \(C^{1, \alpha}\) and \(C^{2, \alpha}\) estimates are presented in Chapter 8. In the final Chapter 9, a \(C^{2, \alpha}\) estimate up to the boundary is proved for the Dirichlet problem (2) when \(F\) is concave, and finally the existence of a unique viscosity solution of this problem is shown.
This book provides a self-contained and detailed presentation of the regularity theory for viscosity solutions of fully nonlinear elliptic equations as developed in the last decade. It can be highly recommended to researchers as well as to graduate students who are interested in this area.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations