[Paper Review] Radial symmetry for p-harmonic functions in exterior and punctured domains

Giorgio Poggesi|arXiv (Cornell University)|Jan 22, 2018

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes radial symmetry for $p$-harmonic functions in exterior and punctured domains under Serrin-type overdetermined boundary conditions, using a $P$-function maximum principle, integral identities, the isoperimetric inequality, and a Soap Bubble-type theorem. The key result is that the only domain admitting such a solution is a ball, even without prior smoothness or star-shapedness assumptions on the domain.

ABSTRACT

We prove symmetry for the p-capacitary potential satisfying $$ \\Delta_p u = 0 \\, \\mbox{ in } \\mathbb{R}^N \\setminus \\overline{\\Omega} , \\; u=1 \\, \\mbox{ on } \\Gamma, \\; \\lim_{|x|\ ightarrow \\infty} u(x)=0 , \\; \\; \\; \\; \\; \\; \\; \\; 1<p<N, $$ under Serrin's overdetermined condition $$ | \ abla u| = c \\mbox{ on } \\Gamma. $$ Here $\\Omega$ is any bounded domain on which no a priori assumption is made, and $\\Gamma$ denotes its boundary. Our result improves on a work of Garofalo and Sartori, where the same conclusion was obtained when $\\Omega$ 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 $$ - \\Delta_p u = K \\, \\delta_0 \\, \\mbox{ in } \\Omega , \\, u=c \\, \\mbox{ on } \\Gamma, \\; \\; \\; \\; \\; \\; \\; \\; 1<p<N, $$ $$ | \ abla u| = 1 \\mbox{ on } \\Gamma , $$ in a bounded star-shaped domain $\\Omega$.

Motivation & Objective

• To prove radial symmetry of $p$-harmonic functions in exterior domains under Serrin's overdetermined condition $|\nabla u| = c$ on $\Gamma$, without assuming smoothness or star-shapedness of the domain $\Omega$.
• To extend symmetry results to the critical case $p = N$, where the $p$-capacitary potential behaves logarithmically at infinity.
• To provide a new proof of symmetry for the interior overdetermined problem with a Dirac delta source, using similar tools as in the exterior case.
• To unify and improve upon prior results by removing restrictive assumptions such as $C^{2,\alpha}$ regularity or star-shapedness, while maintaining strong symmetry conclusions.

Proposed method

• Uses a $P$-function defined as $P = u^{-\frac{p(N-1)}{N-p}} |\nabla u|^p$ to apply the strong maximum principle in $\mathbb{R}^N \setminus \overline{\Omega}$.
• Employs integral identities and the isoperimetric inequality to compare the $L^p$-norm of the gradient on the boundary with geometric quantities.
• Applies a Soap Bubble-type theorem (Theorem A) to deduce that constant mean curvature and equality in the isoperimetric inequality imply spherical symmetry.
• Relies on weak solutions in $W^{1,p}_{\text{loc}}(\mathbb{R}^N \setminus \overline{\Omega})$ and weak boundary conditions to avoid strong regularity assumptions.
• Uses asymptotic expansion of the solution near the origin in the interior problem to relate the constant $K$ in the delta source to surface measure.
• Applies Minkowski's identity and the equality case in the isoperimetric inequality to conclude that mean curvature must be constant, implying spherical symmetry.

Experimental results

Research questions

• RQ1 Under what conditions does the overdetermined $p$-capacitary problem in an exterior domain imply that the domain is a ball?
• RQ2 Can radial symmetry be established for $p$-harmonic functions in exterior domains without assuming the domain is star-shaped or $C^{2,\alpha}$?
• RQ3 How does the symmetry result extend to the critical case $p = N$, where the potential has logarithmic growth?
• RQ4 Can the same method be applied to the interior overdetermined problem with a Dirac delta source to prove symmetry?
• RQ5 What role do integral identities and the $P$-function maximum principle play in proving symmetry without strong regularity assumptions?

Key findings

• For $1 < p < N$, the overdetermined problem $\Delta_p u = 0$ in $\mathbb{R}^N \setminus \overline{\Omega}$, $u = 1$ on $\Gamma$, $\lim_{|x|\to\infty} u = 0$, and $|\nabla u| = c$ on $\Gamma$ admits a weak solution if and only if $\Omega$ is a ball.
• The result holds without any a priori assumption on the regularity or geometry of $\Omega$, improving on previous results that required star-shapedness.
• In the critical case $p = N$, the same symmetry conclusion holds for the problem $\Delta_N u = 0$ in $\mathbb{R}^N \setminus \overline{\Omega}$, $u = 1$ on $\Gamma$, and $u \sim -\ln|x|$ at infinity, under the same overdetermined condition.
• For the interior problem $-\Delta_p u = K\delta_0$ in $\Omega$, $u = c$ on $\Gamma$, $|\nabla u| = 1$ on $\Gamma$, the solution is radially symmetric if $\Omega$ is star-shaped, and $K = |\Gamma|$ is required for consistency.
• The asymptotic behavior of the interior solution near the origin is $u(x) = \frac{p-1}{N-p}\left(\frac{|\Gamma|}{\omega_N}\right)^{\frac{1}{p-1}} |x|^{-\frac{N-p}{p-1}} + o(|x|^{-\frac{N-p}{p-1}})$, which matches the radial solution.
• Equality in the isoperimetric inequality and the mean curvature condition $H \equiv H_0$ imply that $\Gamma$ is a sphere, confirming the domain must be a ball via Alexandrov's Soap Bubble Theorem.