Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

Abstract

We give a complete characterization, as "stadium-like domains", of convex subsets $\Omega$ of ${\mathbb{R}}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $\Omega$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in ${\mathbb{R}}^n$ where the solution to a Dirichlet problem is of class $C^{1,1}(\Omega )$.

21 pages

References in corpus (2)

• On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results
• A $C^1$ regularity result for the inhomogeneous normalized infinity Laplacian

Cited by in corpus (2)

• Rigidity results for variational infinity ground states
• A characterization of cut locus for $C^1$ hypersurfaces