On the viscosity approach to a class of fully nonlinear elliptic equations

Abstract

In this paper, we study some properties of viscosity sub/super-solutions of a class of fully nonlinear elliptic equations relative to the eigenvalues of the complex Hessian. We show that every viscosity subsolution is approximated by a decreasing sequence of smooth subsolutions. When the equations satisfy some conditions on the limit at infinity, we verify that the comparison principle holds, and as a sequence, we obtain a result about the existence of solution of the Dirichlet problem. Using the comparison principle, we show that, under suitable conditions, a Perron-Bremermann envelope can be approximated by a decreasing sequence of viscosity solutions.