Matrix Weights and Regularity for Degenerate Elliptic Equations

TL;DR

This paper establishes local boundedness, Harnack's inequality, and regularity results for solutions of degenerate elliptic equations with rough coefficients, using an axiomatic approach based on geometric conditions and Sobolev inequalities.

Contribution

It introduces a novel axiomatic framework for analyzing degenerate elliptic equations with rough coefficients, extending regularity theory under minimal assumptions.

Findings

Proves local boundedness of solutions.

Establishes Harnack's inequality for degenerate elliptic equations.

Demonstrates local regularity under near-L1 data integrability.

Abstract

We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev-Poincar\'e inequalities. Data integrability is close to L1 and is exploited in terms of a suitable Stummel-Kato class that in some cases is necessary for local regularity.