Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems

Abstract

Let $Ω\subset \mathbb{R}^N$ be a bounded convex domain and let $H_Ω\leq 0$ denote its variational mean curvature, in the sense of Barozzi, Gonzalez and Tamanini [PAMS, 1987]. We prove that $-H_Ω$ is harmonic convex, that is $1/H_Ω$ is a convex function in $Ω$, and that the exponent $-1$ is optimal in the class of convex domains. This strictly improves the concavity of $H_Ω$ obtained by Alter and Caselles [Nonlinear Anal., 2009], who also proved, by geometric methods, a closely related property for the classical mean curvature under Minkowski interpolation of convex bodies. Our approach is analytic: the result is deduced from a concavity property of the solution of the singular $1$-Laplace Dirichlet problem $$\begin{cases} -Δ_1 u = {u^{-γ}} & \quad \hbox{in $Ω$}, \\ u = 0 &\quad \hbox{on $\partial Ω$}, \end{cases}$$ for $γ\in (0,1]$, whose unique positive solution is $u=(-H_Ω)^{-1/γ}$. We show that $u^γ$ is concave, and we obtain this by proving that the solutions $u_p$ of the approximating problems $-Δ_p u_p = u_p^{-γ}$ are $\frac{p-1+γ}{p}$-concave on every bounded convex domain, without any boundary regularity, and then letting $p\to 1^+$. Explicit examples show that no Hölder continuity of $H_Ω$ up to $\partialΩ$ holds uniformly in the class of convex domains with $C^{1,1}$ boundary.