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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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Full Text: Euclid