Optimal concavity of the torsion function

2017/01/20 by Antoine Henrot, Carlo Nitsch, Henrot, A. +5 · 2 citations
Computer Science · Mathematics · #35B06 #35N25 #35R25 #35R30 #52A40 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1701.05821

openalex publication_date 2017/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this short note we consider an unconventional overdetermined problem for the torsion function: let n≥ 2 and Ω be a bounded open set in ℝⁿ whose torsion function u (i.e. the solution to Δu=-1 in Ω, vanishing on ∂Ω) satisfies the following property: √(M-u(x)) is convex, where M=max\u(x) : x∈Ω\. Then Ω is an ellipsoid.

Citations

• A symmetry problem in potential theory
• Symmetry and linear stability in Serrin's overdetermined problem via the\n stability of the parallel surface problem

Cited by

• Hessian degeneracy and non-star-shaped superlevel sets of the torsion function on non-convex domains
• Geometric properties of optimizers for the maximum gradient of the torsion function

Related

• Towards Optimal Gradient Bounds for the Torsion Function in the Plane
• Geometric properties of optimizers for the maximum gradient of the torsion function
• On two functionals involving the maximum of the torsion function
• On convex domains maximizing the gradient of the torsion function
• On capacity and torsional rigidity