Radial symmetry for p-harmonic functions in exterior and punctured domains

AbstractWe prove symmetry for the p-capacitary potential satisfying under Serrin’s overdetermined condition Here is any bounded domain on which no a priori assumption is made, and denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when is star-shaped. Our proof uses the maximum principle for an appropriate P-function, some integral identities, the isoperimetric inequality, and a Soap Bubble-type Theorem. We then treat the case , improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem in a bounded star-shaped domain .