On a Pólya functional for rhombi, isosceles triangles, and thinning convex sets

2018/11/11 by M. van den Berg, Berg, M. van den, Vincenzo Ferone +5 · 1 citation
Computer Science · Mathematics · #35J25 #35P15 #47A75 #49J45 #49R05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1811.04503

openalex publication_date 2018/11/11 · openalex created_date 2018/11/16 · openalex updated_date 2026/07/28

Abstract

Let Ω be an open convex set in \mathbb R^m with finite width, and let v_Ω be the torsion function for Ω, i.e. the solution of -Δv=1, v∈ H₀¹(Ω). An upper bound is obtained for the product of \Vert v_Ω\Vert_L^∞(Ω)λ(Ω), where λ(Ω) is the bottom of the spectrum of the Dirichlet Laplacian acting in L²(Ω). The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area 1, it is shown that \Vert v_Ω\Vert_L¹(Ω)λ(Ω)≥ (π²)/(24), and that this bound is sharp.

Citations

• Hardy inequality and L p estimates for the torsion function

Cited by

• Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

Related

• On a conjecture of a Pólya functional for triangles and rectangles
• On the optimal sets in Pólya and Makai type inequalities
• Spectral bounds for the torsion function
• Sharp and quantitative estimates for the p-Torsion of convex sets
• On two functionals involving the maximum of the torsion function