Geodesic
Boundary-value problem in a~cylinder with frequently changing type of boundary
Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 977-1008

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A singularly perturbed boundary-value problem for the eigenvalues of the Laplace operator in a cylinder with a frequent change of the type of boundary conditions on the lateral surface is considered. The case when the homogenized problem involves the second or the third boundary condition on the lateral surface is studied. For a circular cylinder complete two-parameter asymptotic power series for the eigenvalues and the eigenfunctions of the perturbed problem are constructed. In the case when the section of the cylinder is an arbitrary bounded simply connected domain with smooth boundary, the leading terms of asymptotic formulae for eigenvalues convergent to simple limiting eigenvalues, and the leading terms of asymptotic formulae for the corresponding eigenfunctions are found. 
@article{SM_2002_193_7_a1,
     author = {D. I. Borisov},
     title = {Boundary-value problem in a~cylinder with  frequently changing  type of boundary},
     journal = {Sbornik. Mathematics},
     pages = {977--1008},
     publisher = {mathdoc},
     volume = {193},
     number = {7},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2002_193_7_a1/}
}
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D. I. Borisov. Boundary-value problem in a~cylinder with  frequently changing  type of boundary. Sbornik. Mathematics, Tome 193 (2002) no. 7, pp. 977-1008. http://geodesic.mathdoc.fr/item/SM_2002_193_7_a1/