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 Ω of 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 Ω. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in R n where the solution to a Dirichlet problem is of class C 1,1 (Ω).

A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

“…In the same way it can be shown that it is a subsolution by considering the eigenfunction in balls of radius R contained in Ω. Let us mention that in [9] stadium like domains are characterized by considering a Serrin-type problem for the homogeneous infinity laplacian.…”

A characterization of cut locus for $$C^1$$ hypersurfaces

“…We mention that cut loci also appear in the studies of other partial differential equations, see e.g. [5,6,11].…”

Overdetermined problems for the normalized 𝑝-Laplacian

“…The case p = ∞ was also studied in great detail in a series of papers by Crasta and Fragalá, who relaxed the C 2 smoothness of the boundary, see e.g. [7].…”