Regularity of solutions to degenerate $p$-Laplacian equations

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 Hörmander's ellipticity condition, $A$ is an $m\times m$ symmetric matrix that satisfies degenerate ellipticity conditions. If the degeneracy is of the form \[λw(x)^{2/p}|ξ|^2\leq < A(x)ξ,ξ>\leq Λw(x)^{2/p}|ξ|^2,\] $w \in A_p$, then we show that solutions are locally Hölder continuous. If the degeneracy is of the form \[ k(x)^{-2/p'}|ξ|^2\leq < A(x)ξ,ξ>\leq k(x)^{2/p}|ξ|^2, \] $k\in A_{p'}\cap RH_τ$,where $τ$ 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.

v3 several revisions. Final version. To appear in JMAA