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

Francesca Colasuonno; Fausto Ferrari

doi:10.3934/cpaa.2020045

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2020, Volume 19, Issue 2: 983-1000. Doi: 10.3934/cpaa.2020045

This issue Previous Article Dissipative nonlinear schrödinger equations for large data in one space dimension Next Article Asymptotic behavior of coupled inclusions with variable exponents

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

Francesca Colasuonno^1, , and
Fausto Ferrari^2,

1.
Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, via Carlo Alberto, 10 - 10123 Torino, Italy
2.
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, piazza di Porta S. Donato, 5 - 40126 Bologna, Italy

^* Corresponding author
^* Corresponding author

Received: March 2019

Revised: July 2019

Published: October 2019

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Abstract

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.

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

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References

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