A new notion of Dirichlet boundary condition for elliptic equation of divergence form

Abstract

In this paper, we consider Dirichlet boundary conditions for elliptic equations in divergence form on rough domains by introducing a truncation-based weak solution framework that circumvents global Sobolev trace obstacles. We establish well-posedness on bounded admissible domains for continuous boundary data and $L^2$ sources, deriving boundary Hölder regularities that require only uniform ellipticity and an exterior measure-density condition. This approach eliminates the uniform domain regularity and coefficient continuity imnposed in Guam, Wang and Zhou(2026). Furthermore, on sufficiently flat Reifenberg domains, we obtain boundary $C^{\alpha }$ regularity for any $\alpha \in (0,1)$ using natural energy estimates, thereby bypassing $L^p$($p>\frac{n}{2}$) restrictions found in Lian and Zhang(2026). Crucially, we obtain global $H^1$ regularity for Hölder boundary data ($\alpha >\frac{1}{2}$) a posteriori, without requiring small-BMO coefficient assumptions on the coefficients.