Overdetermined problems for the normalized $p$-Laplacian

TL;DR

This paper extends classical symmetry results to the degenerate game-theoretic p-Laplacian, showing that certain boundary value problems only admit solutions on spherical domains, thus characterizing domain shape through PDE solutions.

Contribution

It generalizes the Serrin and Weinberger symmetry results from the Laplacian to the normalized p-Laplacian, a highly degenerate operator, establishing domain shape constraints for solutions.

Findings

Viscosity solutions exist only on spherical domains.

Extension of symmetry results to degenerate p-Laplacian.

Characterization of domain shape via PDE solutions.

Abstract

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-\Delta_p^Nu=1$ in $\Omega$, $u=0$ and $\tfrac{\partial u}{\partial\nu}=-c\neq 0$ on $\partial\Omega$ can only exist on a bounded domain $\Omega$ if $\Omega$ is a ball.