Boundary pointwise regularity for the divergence form elliptic boundary problem on uniform domain
Tianyu Guan, Lihe Wang, Chunqin Zhou
TL;DR
This work addresses boundary pointwise regularity for divergence-form elliptic boundary problems on $L$-uniform domains with rough boundaries, introducing a weak-solution framework that accommodates nonzero boundary data. It establishes boundary $C^{\alpha}$ regularity via an energy inequality and compactness, and, when boundary data and domain boundary are pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$, derives $C^{1,\alpha}$ and $C^{2,\alpha}$ regularities through harmonic-approximation with a linear/second-order structure. A compactness theory for sequences of $L$-uniform and $\theta$-admissible domains is developed, enabling robust domain perturbation analysis. Collectively, the results extend classical boundary regularity theory to rough domains and nonzero boundary data, with potential future work on the existence theory for such weak solutions.
Abstract
In this paper, we study the boundary pointwise regularity for the divergence form elliptic boundary problem on domains with rough boundaries, specifically uniform domains. In general, it is not straightforward to define weak solutions for non-zero boundary data on such domains. To address this, we introduce a novel definition of weak solutions tailored to the setting of uniform domains. Remarkably, this definition allows for the analysis of the regularity of weak solutions. In particular, by establishing an energy inequality, we prove the boundary pointwise $C^α$ regularity by using compactness methods under the admissible condition. Furthermore, by exploiting the the linear structure of solutions with respective to the harmonic functions, we establish boundary pointwise $C^{1,α}$ and $C^{2,α}$ regularities when the boundary data and the domain boundary are pointwise $C^{1,α}$ and $C^{2,α}$, respectively.