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March/April 2016 Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation
Francoise Demengel
Adv. Differential Equations 21(3/4): 373-400 (March/April 2016).

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.

Citation

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Francoise Demengel. "Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation." Adv. Differential Equations 21 (3/4) 373 - 400, March/April 2016.

Information

Published: March/April 2016
First available in Project Euclid: 18 February 2016

zbMATH: 1339.35132
MathSciNet: MR3461298

Subjects:
Primary: 35B51 , 35B65 , 35D40 , 35D70 , 35J92

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 3/4 • March/April 2016
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