The Dirichlet problem for singular fully nonlinear operators

TL;DR

This paper establishes the existence of viscosity solutions for Dirichlet problems involving singular or degenerate fully nonlinear elliptic operators, extending eigenvalue concepts and analyzing maximum and minimum principles.

Contribution

It introduces a framework for solving Dirichlet problems with singular or degenerate operators and extends eigenvalue theory to this class of problems.

Findings

Existence of viscosity solutions for the class of operators.

Extension of eigenvalue concepts to degenerate and singular operators.

Conditions under which maximum and minimum principles hold.

Abstract

In this paper we prove existence of (viscosity) solutions of Dirichlet problems concerning fully nonlinear elliptic operator, which are either degenerate or singular when the gradient of the solution is zero. For this class of operators it is possible to extend the concept of eigenvalue, this paper concerns the cases when the inf of the principal eigenvalues is positive i.e. when both the maximum and the minimum principle holds.