On higher integrability estimates for elliptic equations with singular coefficients

TL;DR

This paper proves higher integrability and regularity results for solutions to elliptic equations with measurable coefficients, highlighting the importance of the skew-symmetric part of the coefficient matrix and providing counterexamples to common assumptions.

Contribution

It extends classical Meyers' results by including unbounded skew-symmetric parts of the coefficient matrix and demonstrates the necessity of BMO semi-norm dependence for regularity.

Findings

Global reverse Hölder estimates for gradients

Counterexample showing boundedness and ellipticity are insufficient for higher integrability

Dependence of Hölder regularity on BMO semi-norm of skew-symmetric part

Abstract

In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $\text{div}[\mathbf{A}(x) \nabla u] = \text{div}{\mathbf{F}(x)}$, where the matrix $\mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse H\"{o}lder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $\mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $\mathbf{A}$ is the identity matrix. In addition, the example also shows the necessity of the dependence of $\alpha$ in the H\"{o}lder $C^\alpha$-regularity theory on the \textup{BMO}-semi norm of the skew-symmetric part of $\mathbf{A}$. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of $\mathbf{A}$ is assumed to be zero.