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تسجيل مستخدم جديدOn higher integrability estimates for elliptic equations with singular coefficients
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In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $text{div}[mathbf{A}(x) abla u] = text{div}{mathbf{F}(x)}$, where the matrix $mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse H{o}lders 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.
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