February 2020, 19(2): 983-1000. doi: 10.3934/cpaa.2020045
Francesca Colasuonno 1,, and Fausto Ferrari 2,
Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, via Carlo Alberto, 10 - 10123 Torino, Italy
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta S. Donato, 5 - 40126 Bologna, Italy
Received March 2019 Revised July 2019 Published October 2019
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.
Keywords: Alexandrov's soap bubble theorem, Serrin-type result for overdetermined $ p $-Laplacian problems, $ p $-torsional problem, $ P $-function, radial symmetry results.
Mathematics Subject Classification: Primary: 35J92, 35B06, 35N25, 35A23; Secondary: 53A10.
Citation: Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
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Francesca Colasuonno Fausto Ferrari