Geodesic
Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating non-periodic boundary conditions
Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1101-1148.

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We consider a singularly perturbed spectral boundary-value problem for the Laplace operator in a two-dimensional domain with frequently alternating non-periodic boundary conditions. Under certain very weak restrictions on the alternation structure of the boundary conditions, we obtain the first terms of the asymptotic expansions of the eigenelements of this problem. Under still weaker restrictions, we obtain estimates for the rate of convergence of the eigenvalues. 
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D. I. Borisov. Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating non-periodic boundary conditions. Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1101-1148. http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a1/

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