Remarks on regularity for $p$-Laplacian type equations in non-divergence form

TL;DR

This paper investigates the regularity of solutions to a class of singular or degenerate p-Laplacian equations in non-divergence form, establishing local $C^{1,eta}$ regularity and $W^{2,2}$ estimates under certain conditions.

Contribution

It provides new regularity results for viscosity solutions of p-Laplacian type equations in non-divergence form, covering the full range of degeneracy and singularity parameters.

Findings

Proves local $C^{1,eta}$ regularity for solutions in the full parameter range.

Establishes local $W^{2,2}$ estimates near the case where $p$ is close to 2 and $ ext{γ}$ is close to 0.

Extends regularity theory to singular and degenerate regimes of the p-Laplacian equations.

Abstract

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of viscosity solutions in the full range $\gamma>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $\gamma$ is close to 0.