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On Higher Integrability Estimates for Elliptic Equations with Singular Coefficients by Földes, Juraj; Phan, Tuoc is a scholarly article available to read on EtoBox.

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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.

Author

Földes, Juraj; Phan, Tuoc

Published

2018

Language

EN