Harnack inequality and regularity for degenerate quasilinear elliptic equations

TL;DR

This paper establishes Harnack inequalities and regularity results for solutions to degenerate quasilinear elliptic equations, extending classical theory to cases involving degeneracy governed by strong weights.

Contribution

It introduces new regularity results for degenerate equations with minimal assumptions, including $C^{1,\alpha}$ estimates for non-divergence form equations.

Findings

Proved Harnack inequality for degenerate quasilinear elliptic equations.

Established local regularity results under minimal assumptions.

Achieved $C^{1,\alpha}$ estimates for solutions in non-divergence form.

Abstract

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,\alpha}$ local estimates for solutions of a degenerate equation in non divergence form.