Geodesic
On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 56-70

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A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. Complete asymptotic expansions with respect to a parameter (the length of the small arc) are constructed for an eigenvalue of this problem; the eigenvalue converges to a double eigenvalue of the Neumann problem.
Keywords: Laplace operator; singular perturbation; small parameter; eigenvalue; asymptotics. 
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     title = {On an eigenvalue for the {Laplace} operator in a disk with {Dirichlet} boundary condition on a small part of the boundary in a critical case},
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R. R. Gadyl'shin; S. V. Repjevskij; E. A. Shishkina. On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 56-70. http://geodesic.mathdoc.fr/item/TIMM_2015_21_1_a5/