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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case

Author: José M. Arrieta
Journal: Trans. Amer. Math. Soc. 347 (1995), 3503-3531
MSC: Primary 35P15; Secondary 35J05, 35P20
MathSciNet review: 1297521
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Abstract: We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in $ {\mathbb{R}^2}$. This domain consists of two disjoint domains $ {\Omega ^L}$, $ {\Omega ^R}$ joined by a channel $ {R_\varepsilon }$ of height of the order of the parameter $ \varepsilon $. When an eigenvalue approaches an eigenvalue of the Laplacian in $ {\Omega ^L} \cup {\Omega ^R}$, the order of convergence is $ \varepsilon $, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: $ \varepsilon \left\vert {{\text{ln}}\varepsilon } \right\vert$. We also obtain estimates on the behavior of the eigenfunctions.

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