Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

TL;DR

This paper investigates the local behavior of solutions to nonlinear elliptic and parabolic equations with anisotropic weights, establishing asymptotic properties and local Hölder estimates for weighted p-Laplace and fast diffusion equations.

Contribution

It introduces a framework for analyzing solutions with anisotropic weights, covering degenerate and singular cases, and derives new local regularity results.

Findings

Asymptotic behavior characterized near degenerate or singular points.

Established local Hölder continuity for solutions with anisotropic weights.

Extended regularity theory to weighted p-Laplace and fast diffusion equations.

Abstract

The main purpose of this paper is to capture the asymptotic behavior for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted $p$-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local H\"{o}lder estimates for their solutions in the presence of single power-type weights.