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Ufimsk. Mat. Zh., 2016, Volume 8, Issue 4, Pages 63–89 (Mi ufa351)  

The problem of Steklov type in a half-cylinder with a small cavity

D. B. Davletov, D. V. Kozhevnikov

Bashkir State Pedagogical University, Ufa

Abstract: In the work we consider a Steklov type problem for the Laplace operator in $n$-dimensional cylinder with a small cavity. On the lateral surfaces one of three classic boundary conditions is imposed, the boundary of the cavity is subject to the Dirichlet condition, while on the base of the cylinder we impose the spectral Steklov condition. We prove the convergence theorems for the eigenvalues of this problems as the small parameter, the diameter of the cavity, tends to zero. We construct and justify the complete asymptotic expansions in the small parameter converging both to a simple or a double eigenvalue of the limiting problem, which is the problem without the cavity.

Keywords: half-cylinder, Steklov problem, eigenvalue, singular perturbation, small cavity, convergence, asymptotics.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-31-00066
14-01-97024
The work is supported by RFBR (projects no. 16-31-00066-mol_a, 14-01-97024-r_povolzhe_a).


Full text: PDF file (590 kB)
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English version:
Ufa Mathematical Journal, 2016, 8:4, 62–87 (PDF, 533 kB); https://doi.org/10.13108/2016-8-4-62

Bibliographic databases:

UDC: 517.929.7+517.929.8+517.984
MSC: 35J05, 35J25,47A10, 47A55, 47A75, 47F05
Received: 09.06.2016

Citation: D. B. Davletov, D. V. Kozhevnikov, “The problem of Steklov type in a half-cylinder with a small cavity”, Ufimsk. Mat. Zh., 8:4 (2016), 63–89; Ufa Math. J., 8:4 (2016), 62–87

Citation in format AMSBIB
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\issue 4
\pages 63--89
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