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

TL;DR

This paper explores the properties of viscosity solutions for a class of fully nonlinear elliptic equations involving eigenvalues of the complex Hessian, establishing approximation and comparison principles to solve boundary value problems.

Contribution

It introduces new approximation techniques for viscosity solutions and verifies comparison principles under specific conditions, advancing the understanding of complex Hessian equations.

Findings

Viscosity subsolutions can be approximated by decreasing sequences of smooth subsolutions.

Comparison principle holds under certain conditions on the equations.

Existence of solutions for the Dirichlet problem is established using these principles.

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.