A family of degenerate elliptic operators: maximum principle and its consequences

TL;DR

This paper explores the maximum principle for a class of degenerate elliptic operators, revealing unique phenomena, establishing regularity results, and demonstrating existence of solutions and eigenfunctions in convex domains.

Contribution

It introduces new insights into the maximum principle for degenerate elliptic operators involving Hessian eigenvalues, with regularity and existence results for convex domains.

Findings

Maximum principle validity for degenerate elliptic operators.

Lipschitz regularity and boundary estimates in convex domains.

Existence of solutions and principal eigenfunctions.

Abstract

In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of "k" eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates for problems in convex domains which are intersections of ball of same radius. We prove also that smooth bounded strictly convex domains are part of this class. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.