Equivalent Characterizations and Their Applications of Solvability of $L^p$ Poisson--Robin(-Regularity) Problems on Rough Domains

TL;DR

This paper characterizes the solvability of $L^p$ Poisson--Robin problems on rough domains, establishing equivalences, relationships with classical problems, and sharp ranges of $p$ for Lipschitz domains, extending prior results to Robin boundary conditions.

Contribution

It provides new equivalent characterizations of $L^p$ Poisson--Robin problem solvability and explores their relationships with classical problems, including sharp $p$-ranges for Lipschitz domains.

Findings

Equivalent characterizations of solvability established.

Relationship between Poisson--Robin and classical Robin problems clarified.

Sharp $p$-ranges for solvability on Lipschitz domains proven.

Abstract

Let $n\ge2$, $\Omega\subset\mathbb{R}^n$ be a bounded one-sided chord arc domain, and $p\in(1,\infty)$. In this article, we study the (weak) $L^p$ Poisson--Robin(-regularity) problem for a uniformly elliptic operator $L:=-\mathrm{div}(A\nabla\cdot)$ of divergence form on $\Omega$, which considers weak solutions to the equation $Lu=h-\mathrm{div}\boldsymbol{F}$ in $\Omega$ with the Robin boundary condition $A\nabla u\cdot\boldsymbol{\nu}+\alpha u=\boldsymbol{F}\cdot\boldsymbol{\nu}$ on the boundary $\partial\Omega$ for functions $h$ and $\boldsymbol{F}$ in some tent spaces. Precisely, we establish several equivalent characterizations of the solvability of the (weak) $L^p$ Poisson--Robin(-regularity) problem and clarify the relationship between the $L^p$ Poisson--Robin(-regularity) problem and the classical $L^p$ Robin problem. Moreover, we also give an extrapolation property for the solvability of the classical $L^p$ Robin problem. As applications, we further prove that, for the Laplace operator $-\Delta$ on the bounded Lipschitz domain $\Omega$, the $L^p$ Poisson--Robin and the $L^q$ Poisson--Robin-regularity problems are respectively solvable for $p\in(2-\varepsilon_1,\infty)$ and $q\in(1,2+\varepsilon_2)$, where $\varepsilon_1\in(0,1]$ and $\varepsilon_2\in(0,\infty)$ are constants depending only on $n$ and the Lipschitz constant of $\Omega$ and, moreover, these ranges $(2-\varepsilon_1,\infty)$ of $p$ and $(1,2+\varepsilon_2)$ of $q$ are sharp. The main results in this article are the analogues of the corresponding results of both the $L^p$ Poisson--Dirichlet(-regularity) problem, established by M. Mourgoglou, B. Poggi and X. Tolsa [J. Eur. Math. Soc. 2025], and of the $L^p$ Poisson--Neumann(-regularity) problem, established by J. Feneuil and L. Li [arXiv: 2406.16735], in the Robin case.