Lifeboat Foundation

Given an open bounded subset Ω of $mathbbR^n$$ R n, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_infty u = 1$$ — Δ ∞ u = 1 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 C 1(Ω).