Regularity of solutions to degenerate $p$-Laplacian equations

TL;DR

This paper establishes regularity results, including local Hölder continuity and almost everywhere continuity, for solutions to degenerate p-Laplacian equations under specific degeneracy conditions involving Muckenhoupt weights.

Contribution

It provides new regularity theorems for degenerate p-Laplacian equations with weights satisfying A_p and A_{p'} conditions, extending previous results in the field.

Findings

Solutions are locally Hölder continuous under certain degeneracy conditions.

Solutions are continuous almost everywhere with optimal conditions.

Application to maps of finite distortion demonstrates practical relevance.

Abstract

We prove regularity results for solutions of the equation \[div(< AXu,X u>^{(p-2)/2} AX u) = 0,\] $1<p<\infty$, where $X=(X_1,...,X_m)$ is a family of vector fields satisfying H\"ormander's ellipticity condition, $A$ is an $m\times m$ symmetric matrix that satisfies degenerate ellipticity conditions. If the degeneracy is of the form \[\lambda w(x)^{2/p}|\xi|^2\leq < A(x)\xi,\xi>\leq \Lambda w(x)^{2/p}|\xi|^2,\] $w \in A_p$, then we show that solutions are locally H\"older continuous. If the degeneracy is of the form \[ k(x)^{-2/p'}|\xi|^2\leq < A(x)\xi,\xi>\leq k(x)^{2/p}|\xi|^2, \] $k\in A_{p'}\cap RH_\tau$,where $\tau$ depends on the homogeneous dimension, then the solutions are continuous almost everywhere, and we give examples to show that this is the best result possible. We give an application to maps of finite distortion.