Abstract:
We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.
Citation:
D. I. Borisov, “Asymptotics and estimates of the convergence rate in a three-dimensional boundary-value problem with rapidly alternating boundary conditions”, Sibirsk. Mat. Zh., 45:2 (2004), 274–294; Siberian Math. J., 45:2 (2004), 222–240
\Bibitem{Bor04}
\by D.~I.~Borisov
\paper Asymptotics and estimates of the convergence rate in a three-dimensional boundary-value problem with rapidly alternating boundary conditions
\jour Sibirsk. Mat. Zh.
\yr 2004
\vol 45
\issue 2
\pages 274--294
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\transl
\jour Siberian Math. J.
\yr 2004
\vol 45
\issue 2
\pages 222--240
\crossref{https://doi.org/10.1023/B:SIMJ.0000021279.02604.27}
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Linking options:
https://www.mathnet.ru/eng/smj1068
https://www.mathnet.ru/eng/smj/v45/i2/p274
This publication is cited in the following 12 articles:
D. I. Borisov, “Asymptotic Analysis of Boundary-Value Problems for the Laplace Operator with Frequently Alternating Type of Boundary Conditions”, J Math Sci, 277:6 (2023), 841
D. I. Borisov, M. N. Konyrkulzhaeva, “Operator L2-Estimates for Two-Dimensional Problems with Rapidly Alternating Boundary Conditions”, J Math Sci, 267:3 (2022), 319
D. I. Borisov, “Asimptoticheskii analiz kraevykh zadach dlya operatora Laplasa s chastoi smenoi tipa granichnykh uslovii”, Differentsialnye uravneniya s chastnymi proizvodnymi, SMFN, 67, no. 1, Rossiiskii universitet druzhby narodov, M., 2021, 14–129
Chechkina A.G., D'Apice C., De Maio U., “Rate of Convergence of Eigenvalues to Singularly Perturbed Steklov-Type Problem For Elasticity System”, Appl. Anal., 98:1-2, SI (2019), 32–44
Najar H., “Lifshitz Tails For Quantum Waveguides With Random Boundary Conditions”, Math. Phys. Anal. Geom., 22:3 (2019), 17
A. G. Chechkina, “Homogenization of spectral problems with singular perturbation of the Steklov condition”, Izv. Math., 81:1 (2017), 199–236
A. G. Chechkina, V. A. Sadovnichy, “Degeneration of Steklov–type boundary conditions in one spectral homogenization problem”, Eurasian Math. J., 6:3 (2015), 13–29
V. A. Sadovnichii, A. G. Chechkina, “Ob otsenke sobstvennykh funktsii zadachi tipa Steklova s malym parametrom v sluchae predelnogo vyrozhdeniya spektra”, Ufimsk. matem. zhurn., 3:3 (2011), 127–139
Najar H., Olendski O., “Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs”, J. Phys. A: Math. Theor., 44:30 (2011), 305304
Olendski O., Mikhailovska L., “Theory of a curved planar waveguide with Robin boundary conditions”, Physical Review E, 81:3, Part 2 (2010), 036606
A. G. Chechkina, “Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type”, J Math Sci, 162:3 (2009), 443
D. I. Borisov, “On a problem with nonperiodic frequent alternation of boundary conditions imposed on fast oscillating sets”, Comput. Math. Math. Phys., 46:2 (2006), 271–281
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