Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

Abstract

Let $n\ge 2$ and $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ be a bounded Lipschitz domain. In this article, we establish first-order global regularity estimates in the scale of BMO spaces on $\hat{I}\text{c◯}$ for weak solutions to the second-order elliptic equation $\mathrm{div}(A\nabla u)=\mathrm{div},\mathbf{f}$ in $\hat{I}\text{c◯}$. This is achieved under minimal regularity assumptions on $\hat{I}\text{c◯}$ and the coefficient matrix $A$, utilizing the pointwise multiplier characterization of the BMO space on $\hat{I}\text{c◯}$. As an application, we also obtain global estimates of $\nabla u$ in the Lebesgue space $L^1(\hat{I}\text{c◯})$ when $\mathbf{f}$ belongs to the Hardy space on $\hat{I}\text{c◯}$.

30 pages