Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Abstract

Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain $$\Omega $$ . The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz–Morrey spaces as well as Holder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to $$W^{1,p}_{0}(\Omega )$$ for some $$p>1$$ but not necessarily in $$W^{1,2}_{0}(\Omega )$$ ) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.