Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems
Abstract
Let Ω⊂ ℝN be a bounded convex domain and let HΩ≤ 0 denote its variational mean curvature, in the sense of Barozzi, Gonzalez and Tamanini [PAMS, 1987]. We prove that −HΩ is harmonic convex, that is 1/HΩ 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 HΩ 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 −Δ1 u = u−γ & in $Ω$, u = 0 & on $∂ Ω$, for γ∈ (0,1], whose unique positive solution is u=(−HΩ)−1/γ. We show that uγ is concave, and we obtain this by proving that the solutions up of the approximating problems −Δp up = up−γ are (p−1+γ)/p-concave on every bounded convex domain, without any boundary regularity, and then letting p→ 1+. Explicit examples show that no Hölder continuity of HΩ up to ∂Ω holds uniformly in the class of convex domains with C1,1 boundary.
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BibTeX
@misc{gallo2026optimal,
title = {{Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems}},
author = {Marco Gallo and Francesco Petitta and Marco Squassina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28321},
eprint = {2610.28321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28321},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal power concavity of the variational mean curvature via singular $1$-Laplace problems
AU - Marco Gallo
AU - Francesco Petitta
AU - Marco Squassina
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28321
UR - https://arxiv.org/abs/2610.28321
AB - Let Ω⊂ ℝ^N be a bounded convex domain and let H_Ω≤ 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 −Δ_1 u = u^−γ & in $Ω$, u = 0 & on $∂ Ω$, for γ∈ (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 (p−1+γ)/p-concave on every bounded convex domain, without any boundary regularity, and then letting p→ 1^+. Explicit examples show that no Hölder continuity of H_Ω up to ∂Ω holds uniformly in the class of convex domains with C^1,1 boundary.
ER -