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The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem

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  • We consider the $ p $-Laplacian equation $ -\Delta_p u = 1 $ for $ 1<p<2 $, on a regular bounded domain $ \Omega\subset\mathbb R^N $, with $ N\ge2 $, 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\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.

    Mathematics Subject Classification: Primary: 35J92, 35B06, 35N25, 35A23; Secondary: 53A10.

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