On the relations between principal eigenvalue and torsional rigidity

2019/10/31 by M. van den Berg, Berg, Michiel van den, Giuseppe Buttazzo +3 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1910.14593

openalex publication_date 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of minimising or maximising the quantity λ(Ø)T^q(Ø) on the class of open sets of prescribed Lebesgue measure. Here q>0 is fixed, λ(Ø) denotes the first eigenvalue of the Dirichlet Laplacian on H¹₀(Ø), while T(Ø) is the torsional rigidity of Ø. The optimisation problem above is considered in the class of \it all domains Ø, in the class of \it convex domains Ø, and in the class of \it thin domains. The full Blaschke-Santaló diagram for λ(Ø) and T(Ø) is obtained in dimension one, while for higher dimensions we provide some bounds.

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• Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions
• Geometric properties of optimizers for the maximum gradient of the torsion function
• Shape optimization under width constraint: the Cheeger constant and the torsional rigidity

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