Rigidity results for variational infinity ground states

Abstract

We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $\Omega \subset {\mathbb{R}}^n$. They state that $u$ coincides with a multiple of the distance from the boundary of $\Omega$ if either $|\nabla u|$ is constant on $\partial \Omega$, or $u$ is of class $C^{1,1}$ outside the high ridge of $\Omega$. Consequently, in both cases $\Omega$ can be geometrically characterized as a "stadium-like domain".

15 pages

References in corpus (4)

• On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results
• The \infty eigenvalue problem and a problem of optimal transportation
• A $C^1$ regularity result for the inhomogeneous normalized infinity Laplacian
• Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

Cited by in corpus (4)

• The infinity-Fucik spectrum
• Delta-convex structure of the singular set of distance functions
• Principal eigenvalue problem for infinity Laplacian in metric spaces
• Maximal solutions for the Infinity-eigenvalue problem