2017/01/09 by M. van den Berg, Berg, Michiel van den
Computer Science · Mathematics · #35K20 #58J32 #58J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1701.02172
openalex publication_date 2017/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be an open set in Euclidean space \Rm, m=2,3,..., and let vΩ denote the torsion function for Ω. It is known that vΩ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in \Leb2(Ω), denoted by λ(Ω), is bounded away from 0. It is shown that the previously obtained bound ‖vΩ‖\Leb∞(Ω)λ(Ω)≥ 1 is sharp: for m∈\2,3,...\, and any ε>0 we construct an open, bounded and connected set Ωε⊂ \Rm such that ‖vΩε‖\Leb∞(Ωε) λ(Ωε)<1+ε. An upper bound for vΩ is obtained for planar, convex sets in Euclidean space M=\R2, which is sharp in the limit of elongation. For a complete, non-compact, m-dimensional Riemannian manifold M with non-negative Ricci curvature, and without boundary it is shown that vΩ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in \Leb2(Ω) is bounded away from 0.