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

ABSTRACT

We prove symmetry for the p-capacitary potential satisfying $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{p}u=0\text{in}{\mathbb{R}}^N\backslash \overline{\mathrm{\Omega }},u=1\text{on}\mathrm{\Gamma },\underset{|x|\to \infty }{lim}u(x)=0,1<p<N,\end{array}$$

under Serrin’s overdetermined condition $$|\mathrm{\nabla }u|=c\text{on}\mathrm{\Gamma }.$$

Here $\mathrm{\Omega }$ is any bounded domain on which no a priori assumption is made, and $\mathrm{\Gamma }$ denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when $\mathrm{\Omega }$ 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 $1<p=N$, improving previous results present in the literature. Finally, with analogous tools, we give a new proof of symmetry for the interior overdetermined problem $$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u=K{\delta }_0\text{in}\mathrm{\Omega },u=c\text{on}\mathrm{\Gamma },1<p<N,\end{array}$$ $$\begin{array}{c}\hfill |\mathrm{\nabla }u|=1\text{on}\mathrm{\Gamma },\end{array}$$

in a bounded star-shaped domain $\mathrm{\Omega }$.

KEYWORDS:

• Symmetry
• overdetermined boundary problems
• Serrin’s overdetermination
• p-Laplacian
• p-capacitary potential

AMS Subject Classifications:

• Primary: 35N25
• Secondary: 53A10
• 35A23

Acknowledgements

The author wishes to thank Rolando Magnanini for his constructive criticism and for pointing out the paper [27]. 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 [28,43], and [24].

Notes

No potential conflict of interest was reported by the author.

1 More precisely, in free space the intensity of the electric field on $\mathrm{\Gamma }$ is given by ${|\mathrm{\nabla }u|}_{|\mathrm{\Gamma }}=-u_{\nu }=\frac{\rho }{{\epsilon }_0}$, where $\rho$ is the surface charge density over $\mathrm{\Gamma }$ and ${\epsilon }_0$ is the vacuum permittivity. We ignored the constant ${\epsilon }_0$ to be coherent with the mathematical definition of capacity given in (1.1(1.1) $$\begin{array}{c}\hfill {\text{Cap}}_p(\mathrm{\Omega })=inf\left\{{\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }w|}^p:w\in C_0^{\infty }({\mathbb{R}}^N),w\ge 1\text{in}\mathrm{\Omega }\right\}\end{array}$$(1.1) ).

Additional information

Funding

The paper was partially supported by Gruppo Nazionale Analisi Matematica Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).