Weighted-$W^{1,p}$ estimates for weak solutions of degenerate and singular elliptic equations

TL;DR

This paper establishes global weighted $L^{p}$-estimates for the gradient of solutions to certain degenerate and singular elliptic equations on non-smooth domains, extending classical regularity results under specific coefficient and boundary conditions.

Contribution

It introduces new weighted $W^{1,p}$-estimates for weak solutions of degenerate and singular elliptic equations, including a characterization of coefficients satisfying the smallness condition.

Findings

Weighted gradient estimates are proven under smallness and flatness conditions.

A class of coefficients satisfying the smallness condition is characterized.

A counterexample shows the necessity of the smallness condition.

Abstract

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a weight in a Muckenhoupt class. Under a smallness condition on the mean oscillation of the coefficients with the weight and a Reifenberg flatness condition on the boundary of the domain, we establish a weighted gradient estimate for weak solutions of the equation. A class of degenerate coefficients satisfying the smallness condition is characterized. A counter example to demonstrate the necessity of the smallness condition on the coefficients is given. Our $W^{1,p}$-regularity estimates can be viewed as the Sobolev's counterpart of the H\"{o}lder's regularity estimates established by B. Fabes, C. E. Kenig, and R. P. Serapioni in 1982.