On Higher Integrability Estimates for Elliptic Equations with Singular Coefficients

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 Holder'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 Holder $C^\alpha$-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.