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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a bounded convex domain and let $H_{\mathrm{\Omega }}\le 0$ denote its variational mean curvature, in the sense of Barozzi, Gonzalez and Tamanini [PAMS, 1987]. We prove that $-H_{\mathrm{\Omega }}$ is harmonic convex, that is $1\mathrm{/}{H}_{\mathrm{\Omega }}$ is a convex function in $\mathrm{\Omega }$, and that the exponent $-1$ is optimal in the class of convex domains. This strictly improves the concavity of $H_{\mathrm{\Omega }}$ 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 $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{1}u=u^{-\gamma }& \text{in }\mathrm{\Omega },\\ u=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ for $\gamma \in (0,1]$, whose unique positive solution is $u=(-H_{\mathrm{\Omega }}{)}^{-1\mathrm{/}\gamma }$. We show that $u^{\gamma }$ is concave, and we obtain this by proving that the solutions $u_p$ of the approximating problems $-{\mathrm{\Delta }}_p{u}_p=u_p^{-\gamma }$ are $\frac{p-1+\gamma }{p}$-concave on every bounded convex domain, without any boundary regularity, and then letting $p\to 1^{+}$. Explicit examples show that no Hölder continuity of $H_{\mathrm{\Omega }}$ up to $\mathrm{\partial }\mathrm{\Omega }$ holds uniformly in the class of convex domains with $C^{1,1}$ boundary.