Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation

Abstract

We consider the pseudo-$p$-Laplacian operator: $$ \tilde \Delta_p u = \sum_{i=1}^N \partial_i (|\partial_i u|^{p-2} \partial_i u)=(p-1) \sum_{i=1}^N |\partial_i u|^{p-2} \partial_{ii} u \ \text{ for $p > 2$.} $$ We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball $B_1$ of $\tilde \Delta_p u =(p-1) f$ for $f\in { \mathcal C} (\overline{B_1})$ (resp. $f\in L^\infty(B_1)$). First, the Hölder local regularity for any exponent $\gamma < 1$, recovering in that way a known result about weak solutions. Second, we prove the Lipschitz local regularity.