Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems
Abstract
Let Ω⊂ℝn, n≥3, be a bounded uniformly convex C3,γ domain, where 0<γ<1. We study the limit as the positive Robin parameter α→∞. For the $2$-Hessian torsion problem, let U be the admissible Dirichlet solution. For all sufficiently large α, we obtain a unique admissible Robin solution uα and establish the optimal-order estimate |uα−U|C2,γ(Ω̅) ≤ Cα−1. Moreover, −√(−uα) has a uniform positive lower bound on its Hessian, and hence √(−uα) is strictly concave. For the normalized eigenvalue problem, let (λD,φD) be the normalized negative admissible Dirichlet eigenpair, and assume φD∈ C2,γ(Ω̅). For all sufficiently large α, there is a unique normalized negative admissible Robin eigenpair (λα,φα), which satisfies |φα−φD|C2,γ(Ω̅) +|λα−λD|≤ Cα−1. The Hessian of −log(−φα) also has a uniform positive lower bound. The main analytic tool is a uniform Hölder estimate for linear Robin problems, with an α−1 gain for C2,γ boundary data. A normalized augmented inverse removes the scaling kernel in the eigenvalue problem. A Schur-complement argument combines the C2 convergence with boundary geometry to prove convexity up to the boundary and anisotropic Hessian lower bounds at the scale dist(x,∂Ω)+α−1. The convergence rate O(α−1) for the Robin solution functions is optimal even in C0, as follows from their boundary conditions.
Related
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BibTeX
@misc{shi2026power,
title = {{Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems}},
author = {Shujun Shi and Shenghui Sun and Yuzhou Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00836},
eprint = {2611.00836},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00836},
}
RIS
TY - UNPB
TI - Power Concavity and Log-Concavity for Large-Parameter Robin 2-Hessian Problems
AU - Shujun Shi
AU - Shenghui Sun
AU - Yuzhou Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00836
UR - https://arxiv.org/abs/2611.00836
AB - Let Ω⊂ℝ^n, n≥3, be a bounded uniformly convex C^3,γ domain, where 0<γ<1. We study the limit as the positive Robin parameter α→∞. For the $2$-Hessian torsion problem, let U be the admissible Dirichlet solution. For all sufficiently large α, we obtain a unique admissible Robin solution u_α and establish the optimal-order estimate |u_α−U|_C^2,γ(Ω̅) ≤ Cα^−1. Moreover, −√(−u_α) has a uniform positive lower bound on its Hessian, and hence √(−u_α) is strictly concave. For the normalized eigenvalue problem, let (λ_D,φ_D) be the normalized negative admissible Dirichlet eigenpair, and assume φ_D∈ C^2,γ(Ω̅). For all sufficiently large α, there is a unique normalized negative admissible Robin eigenpair (λ_α,φ_α), which satisfies |φ_α−φ_D|_C^2,γ(Ω̅) +|λ_α−λ_D|≤ Cα^−1. The Hessian of −log(−φ_α) also has a uniform positive lower bound. The main analytic tool is a uniform Hölder estimate for linear Robin problems, with an α^−1 gain for C^2,γ boundary data. A normalized augmented inverse removes the scaling kernel in the eigenvalue problem. A Schur-complement argument combines the C^2 convergence with boundary geometry to prove convexity up to the boundary and anisotropic Hessian lower bounds at the scale dist(x,∂Ω)+α^−1. The convergence rate O(α^−1) for the Robin solution functions is optimal even in C^0, as follows from their boundary conditions.
ER -