Abstract
In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet boundary conditions imposed on a subset which is scaling to a point; by a blow-up analysis for the capacitary potentials, we detect the vanishing order of the Sobolev capacity of such shrinking Dirichlet boundary portion.
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Acknowledgements
The authors acknowledge the support of INdAM and CNRS-PICS project n. PICS08262 entitled “VALeurs propres d’un opérateur Aharonov-Bohm avec pôLE variable-VALABLE”. V. Felli is partially supported by the PRIN 2015 grant “Variational methods, with applications to problems in mathematical physics and geometry”. B. Noris was partially supported by the INdAM-GNAMPA Project 2019 “Il modello di Born-Infeld per l’elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni”.
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Communicated by A. Malchiodi.
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Felli, V., Noris, B. & Ognibene, R. Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region. Calc. Var. 60, 12 (2021). https://doi.org/10.1007/s00526-020-01878-3
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DOI: https://doi.org/10.1007/s00526-020-01878-3