Neumann problem with a discontinuous nonlinearity

TL;DR

This paper proves the existence and uniqueness of weak solutions for a nonlinear elliptic Neumann problem with discontinuous nonlinearity and nonsmooth data, providing key estimates for well-posedness.

Contribution

It establishes existence, uniqueness, and well-posedness results for a nonlinear elliptic problem with discontinuous power nonlinearity and nonsmooth boundary data.

Findings

Existence of weak solutions is proven.

An estimate for well-posedness is derived.

Uniqueness holds when the boundary term is smooth.

Abstract

This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is ``smooth".