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Journal of Convex Analysis 31 (2024), No. 2, 315--358
Copyright Heldermann Verlag 2024

On the Spectrum of Sets Made of Cores and Tubes

Francesca Bianchi
Dip. di Scienze Matematiche, Fisiche e Informatiche, University of Parma, Parma, Italy

Lorenzo Brasco
Dip. di Matematica e Informatica, University of Ferrara, Ferrara, Italy

Roberto Ognibene
Dipartimento di Matematica, University of Pisa, Pisa Italy

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: Eigenvalue estimates, Poincaré inequality, inradius, Makai-Hayman inequality, Palais-Smale sequence, curved waveguide.

MSC: 35P15, 35B38, 58E05.

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