2023 Volume 66 Issue 1 Pages 1-16
In this note we establish existence and uniqueness of weak solutions of the linear elliptic equation div[A(x)∇u] = div F(x), where the matrix A is just measurable and its skew-symmetric part can be unbounded. Global reverse Hölder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing examples, that boundedness and ellipticity of A are not sufficient for higher integrability estimates even when the symmetric part of A is the identity matrix. In addition, the examples also show the necessity of the dependence of α in the Hölder Cα-regularity theory on the BMO semi-norm of the skew-symmetric part of A. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of A is assumed to be zero.
