Some possibly degenerate elliptic problems with measure data and non linearity on the boundary

TL;DR

This paper investigates degenerate elliptic equations with measure data and nonlinear boundary conditions, establishing regularity results and estimates for solutions in complex boundary and degeneracy scenarios.

Contribution

It introduces new regularity results for degenerate elliptic problems with measure data and nonlinear boundary conditions, including optimal regularity in certain cases.

Findings

Proved optimal regularity for elliptic problems with measure data.

Provided regularity estimates for degenerate elliptic equations.

Analyzed boundary nonlinearities in measure-supported problems.

Abstract

The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.