Torsional rigidity in random walk spaces

2023/02/28 by José M. Mazón, Mazon, Jose M., J. Toledo +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2302.14351

Abstract

In this paper we study the (nonlocal) torsional rigidity in the ambient space of random walk spaces. We get the relation of the (nonlocal) torsional rigidity of a set Ω with the spectral m-heat content of Ω, what gives rise to a complete description of the nonlocal torsional rigidity of Ω by using uniquely probability terms involving the set Ω; and recover the first eigenvalue of the nonlocal Laplacian with homogeneous Dirichlet boundary conditions by a limit formula using these probability term. For the random walk in \R^N associated with a non singular kernel, we get a nonlocal version of the Saint-Venant inequality, and, under rescaling we recover the classical Saint-Venant inequality. We study the nonlocal p-torsional rigidity and its relation with the nonlocal Cheeger constants. We also get a nonlocal version of the Pólya-Makai-type inequalities. We relate the torsional rigidity given here for weighted graphs with the torsional rigidity on metric graphs.

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