Robin Problems of Elliptic Equations on Rough Domains: Hölder Regularity, Green's Functions, and Harmonic Measures
Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $Ω\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $σ$ is the surface measure on the boundary of $Ω$, denoted by $\partial Ω$. Let $β$ be a non-negative measurable function on $\partial Ω$ satisfying $β\in L^{q_0}(\partial Ω,σ)~\text{with}~ q_0 \in(\frac{s}{s+2-n},\infty]~\text{and}\ β\ge a_0~\text{on}~E_0\subset \partial Ω, $ where $a_0$ is a given positive constant and $E_0\subset \partial Ω$ is a $σ$-measurable set with $σ(E_0)>0$. In this article, for any $f\in L^p(\partial Ω,σ)$ with $p\in(s/(s+2-n),\infty]$, we obtain the existence and uniqueness, the global Hölder regularity, and the boundary Harnack inequality of the weak solution to the Robin problem $$\begin{cases} -\mathrm{div}(A\nabla u) = 0~~&\text{in}~Ω,\\ A\nabla u\cdot \boldsymbolν+βu = f~~&\text{on}~\partial Ω, \end{cases} $$ where the coefficient matrix $A$ is real-valued, bounded and measurable and satisfies the uniform ellipticity condition and where $\boldsymbolν$ denotes the outward unit normal to $\partialΩ$. Furthermore, we establish the existence, upper bound pointwise estimates, and the Hölder 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 $σ$ 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 $β$ is a given positive constant.