## Lipschitz interior regularity for the viscosity and weak solutions of the pseudo $$p$$-Laplacian equation.(English)Zbl 1339.35132

Author’s abstract: We consider the pseudo-$$p$$-Laplacian operator: $\widetilde{\Delta }_pu = \sum _{i=1}^N \partial _i(| \partial _iu | ^{p-2}\partial _iu)= (p-1) \sum _{i=1}^N | \partial _iu | ^{p-2} \partial _{ii} u$ for $$p>2$$. We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball $$B_1$$ of $$\widetilde{\Delta }_pu=(p-1)f$$ for $$f \in C(\overline{B_1})$$ (resp. $$f\in L^{\infty }(B_1))$$. First, we deal with the Hölder local regularity for any exponent $$\gamma <1$$, recovering in that way a known result about weak solutions. Second, we prove the Lipchitz local regularity.

### MSC:

 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35B51 Comparison principles in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35D40 Viscosity solutions to PDEs 35D30 Weak solutions to PDEs
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