On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results

Given an open bounded subset Ω of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}, which is convex and satisfies an interior sphere condition, we consider the pde -Δ∞u=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${-\Delta_{\infty} u = 1}$$\end{document} in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C1(Ω). We then investigate the overdetermined Serrin-type problem, formerly considered in Buttazzo and Kawohl (Int Math Res Not, pp 237–247, 2011), obtained by adding the extra boundary condition |∇u|=a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${|\nabla u| = a}$$\end{document} on ∂Ω; by using a suitable P-function we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball which touches ∂Ω 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 Ω coincide. In turn, in dimension n = 2, this entails that Ω must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class C2.