Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case
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- by José M. Arrieta PDF
- Trans. Amer. Math. Soc. 347 (1995), 3503-3531 Request permission
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: $\varepsilon \left | {{\text {ln}}\varepsilon } \right |$. We also obtain estimates on the behavior of the eigenfunctions.References
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3503-3531
- MSC: Primary 35P15; Secondary 35J05, 35P20
- DOI: https://doi.org/10.1090/S0002-9947-1995-1297521-1
- MathSciNet review: 1297521