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

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.

arXiv comment38 pages