Mixed boundary value problems for fully nonlinear degenerate or singular equations

TL;DR

This paper establishes existence, uniqueness, and regularity for mixed boundary value problems involving fully nonlinear, possibly degenerate or singular elliptic equations, providing a key global H"older estimate.

Contribution

It introduces a novel global H"older estimate for solutions, facilitating analysis of eigenvalues and eigenfunctions in complex boundary value problems.

Findings

Proves existence and uniqueness of solutions.

Derives a global H"older regularity estimate.

Provides a compactness result for solution space.

Abstract

We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global H\"older estimate for solutions, obtained by means of the comparison principle and the construction of ad hoc barriers. The global H\"older estimate immediately yields a compactness result in the space of solutions, which could be applied in the study of principal eigenvalues and principal eigenfunctions of mixed boundary value problems.