Calculus of Variations and Geometric Measure Theory

F. Bianchi - L. Brasco - R. Ognibene

On the spectrum of sets made of cores and tubes

created by brasco on 29 Jun 2023
modified on 15 Jan 2024


Accepted Paper

Inserted: 29 jun 2023
Last Updated: 15 jan 2024

Journal: J. Convex Anal.
Pages: 42
Year: 2023

Dedicated to Giuseppe Buttazzo, a master of Calculus of Variations


We analyze the spectral properties of a particular class of unbounded open sets. These are made of a central bounded ''core'', with finitely many unbounded tubes attached to it. We adopt an elementary and purely variational point of view, studying the compactness (or the defect of compactness) of level sets of the relevant constrained Dirichlet integral. As a byproduct of our argument, we also get exponential decay at infinity of variational eigenfunctions. Our analysis includes as a particular case a planar set (sometimes called ''bookcover''), already encountered in the literature on curved quantum waveguides. J. Hersch suggested that this set could provide the sharp constant in the Makai-Hayman inequality for the bottom of the spectrum of the Dirichlet-Laplacian of planar simply connected sets. We disprove this fact, by means of a singular perturbation technique.

Keywords: Poincare inequality, Inradius, eigenvalue estimates, Makai-Hayman inequality, Palais-Smale sequence, curved waveguide