Boundary pointwise regularity for the divergence form elliptic boundary problem on uniform domain

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 $C^\alpha$ 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 $C^{1,\alpha}$ and $C^{2,\alpha}$ regularities when the boundary data and the domain boundary are pointwise $C^{1,\alpha}$ and $C^{2,\alpha }$, respectively.