Geodesic
Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip
Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 236-252

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In this paper, the method of matching asymptotic expansions is used to construct an asymptotics (in a small parameter) of the eigenvalues and eigenfunctions of the Laplace operator in a domain when the boundary-condition type changes on a narrow flattened strip, provided that on the narrow strip of the boundary a Neumann condition is given and on the remaining part of the boundary a Dirichlet condition is given. The width of the strip is taken as the small parameter. 
@article{MZM_2004_75_2_a6,
     author = {M. Yu. Planida},
     title = {Asymptotics of the {Eigenelements} of the {Laplace} {Operator} when the {Boundary-Condition} {Type} {Changes} on a {Narrow} {Flattened} {Strip}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {236--252},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a6/}
}
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M. Yu. Planida. Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip. Matematičeskie zametki, Tome 75 (2004) no. 2, pp. 236-252. http://geodesic.mathdoc.fr/item/MZM_2004_75_2_a6/