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

Francoise Demengel

#### 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.

#### Article information

Source
Adv. Differential Equations Volume 21, Number 3/4 (2016), 373-400.

Dates
First available in Project Euclid: 18 February 2016

Demengel, Francoise. Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation. Adv. Differential Equations 21 (2016), no. 3/4, 373--400. https://projecteuclid.org/euclid.ade/1455805262