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

Abstract: In this paper, we study the boundary pointwise regularity for the divergence form elliptic boundary problem. In generality, it is not convenient to define weak solutions for nonzero boundary data on domain with the rough boundary, e.g. uniform domain. However, in this paper, we introduce a definition of weak solutions for the boundary problem on uniform domain. What is interesting is that this definition can be considered to analysis the regularity of weak solutions. In particular, by establishing the energy inequality, we show the boundary pointwise $C^\alpha$ regularity by using compactness methods under the admissible condition. Furthermore, by establishing the linear property of solutions with respective to the harmonic functions, we also prove the boundary pointwise $C^{1,\alpha}$ and $C^{2,\alpha}$ regularities if the boundary data and the boundary of domain are pointwise $C^{1,\alpha}$ and $C^{2,\alpha }$ respectively.