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arXiv · 2301.11729

Sharp behavior of Dirichlet--Laplacian eigenvalues for a class of singularly perturbed problems

Laura Abatangelo·Roberto Ognibene
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Abstract

We deepen the study of Dirichlet eigenvalues in bounded domains where a thin tube is attached to the boundary. As its section shrinks to a point, the problem is spectrally stable and we quantitatively investigate the rate of convergence of the perturbed eigenvalues. We detect the proper quantity which sharply measures the perturbation's magnitude. It is a sort of torsional rigidity of the tube's section relative to the domain. This allows us to sharply describe the asymptotic behavior of the perturbed spectrum, even when eigenvalues converge to a multiple one. The final asymptotics of eigenbranches depend on the local behavior near the junction of eigenfunctions chosen in a special way. The present techniques also apply when the perturbation of the Dirichlet eigenvalue problem consists in prescribing homogeneous Neumann boundary conditions on a small portion of the boundary of the domain.

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Laura Abatangelo, Roberto Ognibene. 2023-01-27. Sharp behavior of Dirichlet--Laplacian eigenvalues for a class of singularly perturbed problems. https://arxiv.org/abs/2301.11729

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