On the Asymptotic Behavior of a Simple Eigenvalue of the Steklov Problem Perturbed on a Small Part of the Boundary by the Homogeneous Dirichlet Condition
https://doi.org/10.1134/S2304487X19010073
Abstract
The Steklov type spectral problem for the Laplace operator and the corresponding boundary value problem in a bounded domain with a smooth boundary has been considered. It is assumed that the homogeneous Dirichlet condition is set on a small part of the boundary, and the Steklov condition (or the corresponding Neumann condition) is imposed on the rest of the boundary. It is known that the Steklov problem perturbed on a small part of the boundary by the Dirichlet condition has a countable set of eigenvalues with finite multiplicity. Moreover, the limit problem is a problem for the Laplace operator with the Steklov condition on the entire boundary. It is also known that the problem for the Laplace operator with the Steklov condition on the entire boundary has a countable set of eigenvalues with finite multiplicity. A two-term asymptotic expressions have been constructed for the eigenvalues and the corresponding eigenfunctions of the original problem as the small parameter characterizing the size of the boundary part with the Dirichlet condition tends to zero. It has been shown that the asymptotic expression for the eigenvalue has the second term inversely proportional to the logarithm of the small parameter. Moreover, the asymptotic expression is strictly justified with the estimate of the rest term inversely proportional to the square of the logarithm of the small parameter.
About the Authors
Yu. O. Koroleva
Gubkin Russian State University of Oil and Gas
Russian Federation
Russian Federation
Department of Higher Mathematics
119991
Moscow
T. P. Chechkina
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
References
1. Stekloff, W. Sur les problèmes fondamentaux de la physique mathématique. (French) Ann. Sci. Ecole Norm. Sup., 1902, vol. (3) 19, p. 191–259, 455–490.
2. Mel'nyk T. A. Asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem in a thick periodic junction // Nonlinear Oscillations. 2001. V. 4. № 1. P. 91–105.
3. Pérez E. On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem // Discrete Contin. Dyn. Syst. Ser. B. 2007. V. 7. № 4. P. 859–883.
4. Nazarov S. A., Taskinen J. On the spectrum of the Steklov problem in a domain with a peak. Vestnik St. Petersburg University: Mathematics, 2008, vol. 41, no. 1, p. 45–52.
5. Nazarov S. A. On the spectrum of the Steklov problem in peak-shaped domains. Proceedings of the St. Petersburg Mathematical Society, 2008, vol. XIV, p. 103–168.
6. Nazarov S. A. Asymptotics of the Solution of the Steklov Spectral Problem in a Domain with a Blunted Peak. Math. Notes, 2009, vol. 86, no. 4, p. 542–555.
7. Chechkina A. G. Convergence of Solutions and Eigenelements of Steklov Type Boundary Value Problems with Boundary Conditions of Rapidly Varying Type. Journal of Mathematical Science, 2009, vol. 162, no. 3, p. 443–458.
8. Chechkina A. G. On Singular Perturbations af a Steklov-Type Problem with Asymptotically Degenerate Spectrum. Doklady Mathematics, 2011, vol. 84, no. 2, p. 695–698.
9. Chechkina A. G. Homogenization of Spectral Problems with Singular Perturbation of the Steklov Condition. Izvestiya: Mathematics, 2017, vol. 81, no. 1, p. 199–236.
10. Chechkina A. G., D’Apice C., De Maio U. Rate of Convergence of Eigenvalues to Singularly Perturbed Steklov–Type Problem for Elasticity System // Applicable Analysis. 2019. V. 98. № 1–2. P. 32–44. Online first doi: 10.1080/00036811.2017.1416104
11. Chiado Piat V., Nazarov S., Piatnitski A. Steklov problems in perforated domains with s coefficient of indefinite sign. // Networks Heter. Media. 2012. V. 7. № 1. P. 151–178.
12. Samarskii A. A. O vlijanii zakreplenija na sobstvennye chastoty zamknutykh oblastej [On the Influence of clamping to Eigenvalues of Closed Domains]. Doklady SSSR. Mathematics. 1948, vol. 63, no. 6, p. 631–634. In Russian.
13. Swanson C. A. Asymptotic Variational Formulae for Eigenvalues // Canad. Math. Bul. 1963. V. 6. № 1. P. 1525.
14. Ozawa Sh. Singular Hadamard’s Variation of Domains and Eigenvalues of Laplacian 1 // Proc. Japan Acad. Ser. A. 1981. V. 57. № 5. P. 242–246.
15. Maz’ya V. G., Nazarov S. A., Plamenevskii B. A., Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes. Math. USSR-Izvestia, 1985, vol. 24, no. 2, p. 321–345.
16. Gadyl’shin R. R. Singuljarno vozmushchennye zadachi dlja operatora Helmholza. Singularly Perturbed problem for the Helmholz Operator. In: Differential Equations with Small Parameter. Sverdlovsk, UrO AN SSSR, 1984, p. 18–35. In Russian.
17. Gadyl’shin R. R. Asymptotic properties of an eigenvalue of a problem for a singularly perturbed self-adjoint elliptic equation with a small parameter in the boundary conditions. Differ. Equations, 1985, vol. 22, p. 474–483.
18. Gadyl’shin R. R. Spektr ellipticheskoj kraevoj zadachi s singuljarnym vozmushcheniem kraevykh uslovij Spectrum of Elliptic Boundary Value Problems with Singular Perturbation of Boundary Conditions. In: Asymptotic Properties of Solutions to Differential Equations. Ufa. BNC UrO AN SSSR, 1988, p. 3–15. In Russian.
19. Il’in A. M. Matching of asymptotic expansions of solutions of boundary value problems, Translations of Math. Monographs 102, AMS, Providence, 1992.
20. Gadyl’shin R. R., Koroleva Yu. O., Chechkin G. A. On the eigenvalue of the Laplacian in a domain perforated along the boundary. Doklady Mathematics, 2010, vol. 81, no. 3, p. 337–341.
21. Gadyl’shin R. R. Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary. Sib. Math. J., 1993, vol. 34, no. 3, p. 433–450.
22. Gadyl’shin R. R., Piatnitski A. L., Chechkin G. A. On Asymptotic Behavior of Eigenelements to a Boundary Value Problem in Plane Domain of the Steklov Sieve Type. Izvestia RAS, 2018, vol. 82, no. 6, p. 37–64.
23. Abdullazade N. N., Chechkin G. A. On Perturbation of a Steklov problem on a Small Part of the Boundary. Journal of Mathematical Sciences, 2014, vol. 196, no. 4, p. 441–451.
24. Chechkin G. A., D’Apice C., De Maio U., Gadyl’shin R. R. On Singularly Perturbed Steklov problem in Domain Perforated Along the Boundary // C R Mécanique. 2016. Vol. 344. № 1. P. 12–18.
25. Borisov D. I. Boundary-value problem in a cylinder with frequently changing type of boundary. Sb. Math, 2002, vol. 193, no. 7, p. 977–1008.
26. Borisov D. I. On a model boundary value problem for Laplacian with frequently alternating type of boundary condition // Asymptotic Analysis. 2003. V. 35. № 1. P. 1–26.
27. Borisov D. I. O RT-simmetrichnom volnovode s paroj malen’kikh otverstij On RT-symmetric waveguide with couple of small holes. Proceedings of the Institute of Mathematics and Mechanics UrO RAS, 2012, vol. 18, no. 2, p. 22–37. In Russian.
28. Borisov D., Cardone G., Durante T. Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve // Proceedings of the Royal Society of Edinburgh, Section: A Mathematics. 2016. V. 146. № 6. P. 1115–1158.
29. Borisov D. Creation of spectral bands for a periodic domain with small windows // Russian Journal of Mathematical Physics. 2016. V. 23. № 1. P. 19–34.
30. Borisov D. I., Mukhametrakhimova A. I. On norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes. Journal of Mathematical Sciences, 2018, vol. 232, no. 3, p. 283–298.
31. Chechkin G. A. On Vibration of Partially Fastened Membrane with Many “light” Concentrated Masses on the Boundary // C R Mécanique. 2004. Vol. 332. № 12. P. 949–954.
32. Amirat Y., Chechkin G. A., Gadyl’shin R. R. Asymptotics of Simple Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary. Computational Mathematics and Mathematical Physics, 2006, vol. 46, no. 1, p. 97–110.
33. Mel’nyk T. A., Chechkin G. A. Homogenization of a Boundary-Value Problem in a Thick 3-Dimensional Multilevel Junction. Russian Academy of Sciences. Sbornik. Mathematics, 2009, vol. 200, no. 3, p. 357–383.
34. Amirat Y., Chechkin G. A., Gadyl’shin R. R. Asymptotics of the Solution of a Dirichlet Spectral Problem in a Junction with Highly Oscillating Boundary // C R Mécanique. 2008. Vol. 336. № 9. P. 693–698.
35. Il’in A. M., A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case. Math. USSR-Sb., 1976, vol. 28, no. 4, p. 459–480.
36. Il’in A. M., A boundary value problem for the second order elliptic equation in a domain with a narrow slit. 2. Domain with a small cavity. Math. USSR-Sb., 1977, vol. 32, no. 2, p. 227–244.
37. Il’in A. M. Issledovanie asimptotiki reshenij ellipticheskoj kraevoj zadachi v oblasti s malym otverstiem An investigation of the asymptotics for a solution of an elliptic boundary value problem in a domain with a small hole. Trudy Semin. I. G. Petrovskogo, 1981, vol. 6, p. 57–82. In Russian.
38. Chechkin G. A. Asymptotic Expansion of Eigenvalues and Eigenfunctions of an Elliptic Operator in a Domain with Many “Light” Concentrated Masses Situated on the Boundary. Two-Dimensional Case. // Izvestia: Mathematics. 2005. Vol. 69. № 4. P. 805–846 (Translated from Izvestija RAN. Ser. Mat. 2005. Vol. 69. № 4. P. 161–204).
39. Borisov D. I. Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating nonperiodic boundary conditions. Izvestia: Mathematics, 2003, vol. 67, no. 6, p. 1101–1148.
Review
For citations:
Koroleva Yu.O., Chechkina T.P. On the Asymptotic Behavior of a Simple Eigenvalue of the Steklov Problem Perturbed on a Small Part of the Boundary by the Homogeneous Dirichlet Condition. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(3):253-263. (In Russ.) https://doi.org/10.1134/S2304487X19010073
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