Gradient weighted norm inequalities for linear elliptic equations with discontinuous coefficients

TL;DR

This paper establishes weighted norm inequalities for the gradient of solutions to linear elliptic equations with discontinuous coefficients, applicable to very weak solutions in Lipschitz domains, extending classical estimates.

Contribution

It introduces new weighted gradient estimates for elliptic equations with discontinuous coefficients in Lipschitz domains, including very weak solutions, using localized maximal function techniques.

Findings

Weighted gradient estimates in Lebesgue and Lorentz spaces

Extension to very weak solutions with discontinuous coefficients

Optimal estimates matching classical results in smooth cases

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 H\"older 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.