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A boundary value problem for the Laplacian with rapidly changing type of boundary conditions in a multidimensional domain. (English. Russian original) Zbl 0939.35048

Sib. Math. J. 40, No. 2, 229-244 (1999); translation from Sib. Mat. Zh. 40, No. 2, 271-287 (1999).
In the article under review, the authors study boundary value problems for the Laplacian in \(\mathbb R^n\), \(n\leq 3\), with rapidly changing boundary conditions. As in the case of \(n=2\) considered in [G. A. Chechkin, Mat. Sb., 184, No. 6, 99-150 (1993; Zbl 0875.35009)], the explicit form of the limit problem depends on the asymptotic behavior of the first eigenvalue of the spectral problem for the Laplacian operator that is cellular. An essential part of the article is devoted to studying the indicated asymptotic behavior for \(n\geq 3\). The authors use the methods worked out by Oleĭnik.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs

Citations:

Zbl 0875.35009
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