We consider the p-Laplacian equation -Delta(p)u = 1 for 1 < p < 2, on a regular bounded domain Omega subset of R-N, with N >= 2, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature H of partial derivative Omega is constant, then Omega is a ball and the unique solution of the Dirichlet p-Laplacian problem is radial. The main tools used are integral identities, the P-function, and the maximum principle.
The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem / Colasuonno, Francesca; Ferrari, Fausto. - In: COMMUNICATIONS ON PURE AND APPLIED ANALYSIS. - ISSN 1534-0392. - STAMPA. - 19:2(2020), pp. 983-1000. [10.3934/cpaa.2020045]
The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem
Colasuonno, Francesca;Ferrari, Fausto
2020
Abstract
We consider the p-Laplacian equation -Delta(p)u = 1 for 1 < p < 2, on a regular bounded domain Omega subset of R-N, with N >= 2, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature H of partial derivative Omega is constant, then Omega is a ball and the unique solution of the Dirichlet p-Laplacian problem is radial. The main tools used are integral identities, the P-function, and the maximum principle.File | Dimensione | Formato | |
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