The spectral gap to torsion problem for some non-convex domains

2021/02/09 by Hua Chen, Peng Luo, Chen, Hua +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2102.04900

openalex publication_date 2021/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the following torsion problem \begincases -Δu=1~amp;in Ω,
u=0~amp;on ∂Ω. \endcases Let Ω⊂ ℝ² be a bounded, convex domain and u₀(x) be the solution of above problem with its maximum y₀∈ Ω. Steinerberger proved that there are universal constants c₁, c₂>0 satisfying λ_max(D²u₀(y₀))≤ -c₁exp(-c₂(diam(Ω))/(inrad(Ω))). And he proposed following open problem: "Does above result hold true on domains that are not convex but merely simply connected or perhaps only bounded? The proof uses convexity of the domain Ω in a very essential way and it is not clear to us whether the statement remains valid in other settings." Here by some new idea involving the computations on Green's function, we compute the spectral gap λ_maxD²u(y₀) for some non-convex smooth bounded domains, which gives a negative answer to above open problem. Also some extensions are given.

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• Hessian degeneracy and non-star-shaped superlevel sets of the torsion function on non-convex domains

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