Geodesic
Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets
Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 703-741

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Perturbations of the three-dimensional Dirichlet problem in a bounded domain are studied. One type of perturbation is change of type of the boundary condition on a narrow strip contracting to a closed curve on the boundary. The second type of perturbation is effected by cutting out in the domain a thin ‘toroidal’ body, also contracting to a closed curve (but now contained inside the domain) and imposing a Neumann boundary condition at the boundary of this thin body. For these problems the method of matched asymptotic expansions is used to construct complete asymptotics (in a small parameter) of the eigenvalues, converging to the simple eigenvalues of the unperturbed problem, and of the corresponding eigenfunctions. The small parameter is the width of the strip and the diameter of a section of the torus, respectively. 
@article{SM_2005_196_5_a3,
     author = {M. Yu. Planida},
     title = {Asymptotics of the eigenelements of the {Laplacian} with singular perturbations of boundary conditions on narrow and thin sets},
     journal = {Sbornik. Mathematics},
     pages = {703--741},
     publisher = {mathdoc},
     volume = {196},
     number = {5},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_5_a3/}
}
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M. Yu. Planida. Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets. Sbornik. Mathematics, Tome 196 (2005) no. 5, pp. 703-741. http://geodesic.mathdoc.fr/item/SM_2005_196_5_a3/