Matrix Weights and Regularity for Degenerate Elliptic Equations

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.