ABSTRACT
We 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 .
AMS Subject Classifications:
Acknowledgements
The author wishes to thank Rolando Magnanini for his constructive criticism and for pointing out the paper [Citation27]. The author also wishes to thank Nicola Garofalo for his kind interest in this work, and Chiara Bianchini, Andrea Colesanti, and Paolo Salani for bringing to his attention the references [Citation28,Citation43], and [Citation24].
Notes
No potential conflict of interest was reported by the author.
1 More precisely, in free space the intensity of the electric field on is given by
, where
is the surface charge density over
and
is the vacuum permittivity. We ignored the constant
to be coherent with the mathematical definition of capacity given in (Equation1.1
(1.1)
(1.1) ).