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

Abstract

We introduce the $L^p$ Poisson-Neumann problem for an uniformly elliptic operator $L=-\mathrm{div}A\nabla$ in divergence form in a bounded 1-sided Chord Arc Domain $\hat{I}\text{c◯}$, which considers solutions to $Lu=h-\mathrm{div}\vec{\mathrm{F}}$ in $\hat{I}\text{c◯}$ 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 Hölder 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.

49 pages