标题： 椭圆方程在粗糙区域上的罗宾问题：霍尔德正则性、格林函数和调和测度 显示英文标题

标题： Robin Problems of Elliptic Equations on Rough Domains: Hölder Regularity, Green's Functions, and Harmonic Measures

摘要： 设 $n\ge 2$ 和 $s\in (n-2,n)$。 假设 $\Omega\subset \mathbb{R}^n$是一个单侧有界非切向可访问域，其边界为 $s$-Ahlfors 正则，并且 $\sigma$是边界上的表面测度，记为 $\partial \Omega$，该边界由 $\Omega$表示。 设 $\beta$ 是 $\partial \Omega$ 上的一个非负可测函数，满足 $\beta\in L^{q_0}(\partial \Omega,\sigma)~\text{with}~ q_0 \in(\frac{s}{s+2-n},\infty]~\text{and}\ \beta\ge a_0~\text{on}~E_0\subset \partial \Omega, $，其中 $a_0$ 是一个给定的正常数，而 $E_0\subset \partial \Omega$ 是一个 $\sigma$-可测集，且 $\sigma(E_0)>0$。 在本文中，对于任何具有$p\in(s/(s+2-n),\infty]$的$f\in L^p(\partial \Omega,\sigma)$，我们得到 Robin 问题$$\begin{cases} -\mathrm{div}(A\nabla u) = 0~~&\text{in}~\Omega,\\ A\nabla u\cdot \boldsymbol{\nu}+\beta u = f~~&\text{on}~\partial \Omega, \end{cases} $$弱解的存在性和唯一性，全局 Hölder 正则性以及边界 Harnack 不等式，其中系数矩阵$A$是实值、有界可测的，并满足一致椭圆性条件，其中$\boldsymbol{\nu}$表示到$\partial\Omega$的外单位法向量。 此外，我们建立了与该 Robin 问题相关的 Green 函数的存在性、逐点估计的上界以及 Hölder 正则性。 作为应用，我们进一步证明了与该 Robin 问题相关的调和测度与表面测度$\sigma$是相互绝对连续的，并且还提供了在小尺度下相互绝对连续性的定量特征。 这些结果扩展了 David 等人建立的相应结果。 [arXiv: 2410.23914] 通过削弱他们假设$\beta$是一个给定的正数。 显示英文摘要

摘要： Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $\Omega\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $\sigma$ is the surface measure on the boundary of $\Omega$, denoted by $\partial \Omega$. Let $\beta$ be a non-negative measurable function on $\partial \Omega$ satisfying $\beta\in L^{q_0}(\partial \Omega,\sigma)~\text{with}~ q_0 \in(\frac{s}{s+2-n},\infty]~\text{and}\ \beta\ge a_0~\text{on}~E_0\subset \partial \Omega, $ where $a_0$ is a given positive constant and $E_0\subset \partial \Omega$ is a $\sigma$-measurable set with $\sigma(E_0)>0$. In this article, for any $f\in L^p(\partial \Omega,\sigma)$ with $p\in(s/(s+2-n),\infty]$, we obtain the existence and uniqueness, the global H\"older regularity, and the boundary Harnack inequality of the weak solution to the Robin problem $$\begin{cases} -\mathrm{div}(A\nabla u) = 0~~&\text{in}~\Omega,\\ A\nabla u\cdot \boldsymbol{\nu}+\beta u = f~~&\text{on}~\partial \Omega, \end{cases} $$ where the coefficient matrix $A$ is real-valued, bounded and measurable and satisfies the uniform ellipticity condition and where $\boldsymbol{\nu}$ denotes the outward unit normal to $\partial\Omega$. Furthermore, we establish the existence, upper bound pointwise estimates, and the H\"older regularity of Green's functions associated with this Robin problem. As applications, we further prove that the harmonic measure associated with this Robin problem is mutually absolutely continuous with respect to the surface measure $\sigma$ and also provide a quantitative characterization of mutual absolute continuity at small scales. These results extend the corresponding results established by David et al. [arXiv: 2410.23914] via weakening their assumption that $\beta$ is a given positive constant.