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

TL;DR

This paper proves global BMO space estimates for solutions to second-order elliptic equations on Lipschitz domains, under minimal regularity assumptions, and extends results to $L^1$ estimates when data is in Hardy spaces.

Contribution

It establishes the first-order global BMO regularity estimates for elliptic equations on Lipschitz domains with minimal regularity assumptions.

Findings

Established BMO regularity estimates for weak solutions.

Derived $L^1$ estimates of $ abla u$ for data in Hardy spaces.

Utilized pointwise multiplier characterization of BMO spaces.

Abstract

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