The $L^p$ Poisson-Neumann problem and its relation to the Neumann problem

TL;DR

This paper introduces the $L^p$ Poisson-Neumann problem for elliptic operators in certain domains, characterizes its solvability, and relates it to the longstanding open problem of $L^p$ Neumann problem solvability, improving previous extrapolation results.

Contribution

It defines the $L^p$ Poisson-Neumann problem, provides solvability characterizations, and links it to the $L^p$ Neumann problem, advancing understanding and extrapolation techniques.

Findings

Solvability characterized by weak reverse H"older inequality

Weak $L^p$ Poisson-Neumann solvability equivalent to certain inequalities

Improved extrapolation results for $L^p$ Neumann problem

Abstract

We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\rm{div }A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $\Omega$, which considers solutions to $Lu=h-\rm{div}\vec{F}$ in $\Omega$ with zero Neumann data on the boundary for $h$ and $\vec F$ in some tent spaces. We give different characterizations of solvability of the $L^p$ Poisson-Neumann problem and its weaker variants, and in particular, we show that solvability of the weak $L^p$ Poisson-Neumann probelm is equivalent to a weak reverse H\"older inequality. We show that the Poisson-Neumman problem is closely related to the $L^p$ Neumann problem, whose solvability is a long-standing open problem. We are able to improve the extrapolation of the $L^p$ Neumann problem from Kenig and Pipher by obtaining an extrapolation result on the Poisson-Neumann problem.