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

Abstract

Let $n\ge2$, $Ω\subset\mathbb{R}^n$ 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 $Ω$, which considers weak solutions to the equation $Lu=h-\mathrm{div}\boldsymbol{F}$ in $Ω$ with the Robin boundary condition $A\nabla u\cdot\boldsymbolν+αu=\boldsymbol{F}\cdot\boldsymbolν$ on the boundary $\partialΩ$ 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 $-Δ$ on the bounded Lipschitz domain $Ω$, 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 $Ω$ 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.