Matrix $A_p$ weights, degenerate Sobolev spaces, and mappings of finite distortion

TL;DR

This paper investigates degenerate Sobolev spaces influenced by matrix $A_p$ weights, establishing a classical approximation theorem and applying it to partial regularity of solutions and mappings of finite distortion.

Contribution

It extends the Meyers-Serrin approximation theorem to matrix-weighted degenerate Sobolev spaces and applies this to PDE regularity and finite distortion mappings.

Findings

Meyers-Serrin theorem holds in matrix-weighted degenerate Sobolev spaces.

Partial regularity results for degenerate p-Laplacian equations.

Applications to mappings of finite distortion.

Abstract

We study degenerate Sobolev spaces where the degeneracy is controlled by a matrix $A_p$ weight. This class of weights was introduced by Nazarov, Treil and Volberg, and degenerate Sobolev spaces with matrix weights have been considered by several authors for their applications to PDEs. We prove that the classical Meyers-Serrin theorem, H = W, holds in this setting. As applications we prove partial regularity results for weak solutions of degenerate p-Laplacian equations, and in particular for mappings of finite distortion.