Existence and regularity results for a Neumann problem with mixed local and nonlocal diffusion

TL;DR

This paper establishes existence and regularity results for an elliptic Neumann problem involving a combined local and nonlocal diffusion operator with drift, under specific conditions on the source term.

Contribution

It provides the first rigorous proof of existence, uniqueness, and regularity for solutions to a mixed local-nonlocal elliptic problem with Neumann boundary conditions.

Findings

Existence and uniqueness of solutions in Sobolev space W^{2,p}

Solutions exist for certain ranges of p and s

Regularity results for the mixed diffusion problem

Abstract

In this paper, we consider an elliptic problem driven by a mixed local-nonlocal operator with drift and subject to nonlocal Neumann condition. We prove the existence and uniqueness of a solution $u\in W^{2,p}(\Omega)$ of the considered problem with $L^p$-source function when $p$ and $s$ are in a certain range.