On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results

Abstract

Given an open bounded subset $\Omega$ of ${\mathbb{R}}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-{\Delta }_{\infty }u=1$ in $\Omega$, subject to the homogeneous boundary condition $u=0$ on $\partial \Omega$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C^1(\Omega )$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u|=a$ on $\partial \Omega$; by using a suitable $P$-function we prove that, if $\Omega$ satisfies the same assumptions as above and in addition contains a ball with touches $\partial \Omega$ at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of $\Omega$ coincide. In turn, in dimension $n=2$, this entails that $\Omega$ must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class $C^2$.

26 pages, 1 figure

References in corpus (1)

• Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

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