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

Abstract

Let $n\ge 2$, $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ 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 $\hat{I}\text{c◯}$, which considers weak solutions to the equation $Lu=h-\mathrm{div}\mathbf{F}$ in $\hat{I}\text{c◯}$ with the Robin boundary condition $A\nabla u\cdot \hat{\mathbf{I}}½+\hat{I}\pm u=\mathbf{F}\cdot \hat{\mathbf{I}}½$ on the boundary $\partial \hat{I}\text{c◯}$ for functions $h$ and $\mathbf{F}$ 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 $-\hat{I}”$ on the bounded Lipschitz domain $\hat{I}\text{c◯}$, the $L^p$ Poisson--Robin and the $L^q$ Poisson--Robin-regularity problems are respectively solvable for $p\in (2-{\epsilon }_1,\infty )$ and $q\in (1,2+{\epsilon }_2)$, where ${\epsilon }_1\in (0,1]$ and ${\epsilon }_2\in (0,\infty )$ are constants depending only on $n$ and the Lipschitz constant of $\hat{I}\text{c◯}$ and, moreover, these ranges $(2-{\epsilon }_1,\infty )$ of $p$ and $(1,2+{\epsilon }_2)$ 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.