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

We prove symmetry for the p-capacitary potential satisfying (Formula presented.) under Serrin’s overdetermined condition (Formula presented.) Here (Formula presented.) is any bounded domain on which no a priori assumption is made, and (Formula presented.) denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when (Formula presented.) 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 (Formula presented.), improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem (Formula presented.) (Formula presented.) in a bounded star-shaped domain (Formula presented.).