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 Mat. Sb., 1993, Volume 184, Number 6, Pages 99–150 (Mi msb995)

This article is cited in 50 scientific papers (total in 50 papers)

Averaging of boundary value problems with a singular perturbation of the boundary conditions

G. A. Chechkin

Abstract: Boundary value problems with different conditions on alternating small parts of the boundary are considered. An investigation is made of the behavior of solutions of such problems as the small parameter characterizing the period of change of the type of the boundary conditions goes to zero, and estimates are given for the deviation of these solutions from the solutions of the limit problem in various cases. The spectral properties of these problems are studied from a unified point of view on the basis of general methods (see [4], [9]).

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 79:1, 191–222

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J25, 35J05; Secondary 73K10, 35B25
Received: 23.04.1992

Citation: G. A. Chechkin, “Averaging of boundary value problems with a singular perturbation of the boundary conditions”, Mat. Sb., 184:6 (1993), 99–150; Russian Acad. Sci. Sb. Math., 79:1 (1994), 191–222

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. A. Oleinik, G. A. Chechkin, “On boundary-value problems for elliptic equations with rapidly changing type of boundary conditions”, Russian Math. Surveys, 48:6 (1993), 173–175
2. A. G. Belyaev, G. A. Chechkin, “Homogenization of a mixed boundary-value problem for the Laplace operator in the case of an insoluble 'limit' problem”, Sb. Math., 186:4 (1995), 511–525
3. Skrypnik I., “Asymptotic Behavior of Solutions to Nonlinear Elliptic Problems with Nonhomogeneous Boundary Conditions”, Differ. Equ., 31:3 (1995), 507–511
4. Friedman A., Huang C., Yong J., “Effective Permeability of the Boundary of a Domain”, Commun. Partial Differ. Equ., 20:1-2 (1995), 59–102
5. R. R. Gadyl'shin, “On the perturbation of the Laplacian spectrum when the boundary condition type changes on a small part of the boundary”, Comput. Math. Math. Phys., 36:7 (1996), 889–898
6. Gadylshin R., “Asymptotics of the Minimum Eigenvalue for a Circle with Fast Oscillating Boundary Conditions”, Comptes Rendus Acad. Sci. Ser. I-Math., 323:3 (1996), 319–323
7. R. R. Gadyl'shin, “Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator”, Russian Math. Surveys, 52:1 (1997), 1–72
8. D. I. Borisov, R. R. Gadyl'shin, “On the spectrum of the Laplacian with frequently alternating boundary conditions”, Theoret. and Math. Phys., 118:3 (1999), 272–277
9. Alexander G. Belyaev, Gregory A. Chechkin, Rustem R. Gadyl'shin, “Effective Membrane Permeability: Estimates and Low Concentration Asymptotics”, SIAM J Appl Math, 60:1 (1999), 84
10. A. Yu. Belyaev, G. A. Chechkin, “Averaging of operators with a fine-scaled structure of boundary conditions”, Math. Notes, 65:4 (1999), 418–429
11. R GADYLSHIN, “On an analog of the Helmholtz resonator in the averaging theory”, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 329:12 (1999), 1121
12. Gadyl'shin R., “Asymptotics of the Eigenvalues of a Boundary Value Problem with Rapidly Oscillating Boundary Conditions”, Differ. Equ., 35:4 (1999), 540–551
13. D. I. Borisov, “Two-Parameter Asymptotics in a Boundary-Value Problem for the Laplacian”, Math. Notes, 70:4 (2001), 471–485
14. R. R. Gadyl'shin, “Homogenization and asymptotics for a membrane with closely spaced clamping points”, Comput. Math. Math. Phys., 41:12 (2001), 1765–1776
15. Mel'nyk T., “Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator”, Z. Anal. ihre. Anwend., 20:4 (2001), 941–957
16. R. R. Gadyl'shin, “On a Model Analogue of the Helmholtz Resonator in Homogenization”, Proc. Steklov Inst. Math., 236 (2002), 70–77
17. D. I. Borisov, “Boundary-value problem in a cylinder with frequently changing type of boundary”, Sb. Math., 193:7 (2002), 977–1008
18. R. R. Gadyl'shin, “Analogues of the Helmholtz resonator in homogenization theory”, Sb. Math., 193:11 (2002), 1611–1638
19. Borisov D., “On a Singularly Perturbed Boundary Value Problem for the Laplacian in a Cylinder”, Differ. Equ., 38:8 (2002), 1140–1148
20. Borisov D., “On a Laplacian with Frequently Nonperiodically Alternating Boundary Conditions”, Dokl. Math., 65:2 (2002), 224–226
21. D. I. Borisov, “Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating non-periodic boundary conditions”, Izv. Math., 67:6 (2003), 1101–1148
22. D. I. Borisov, “Asymptotics and estimates of the convergence rate in a three-dimensional boundary-value problem with rapidly alternating boundary conditions”, Siberian Math. J., 45:2 (2004), 222–240
23. G. A. Chechkin, “Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses”, Math. Notes, 76:6 (2004), 865–879
24. G. A. Chechkin, “Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case”, Izv. Math., 69:4 (2005), 805–846
25. M. Yu. Planida, “Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets”, Sb. Math., 196:5 (2005), 703–741
26. Chechkin G., Gadyl'shin R., “On Boundary-Value Problems for the Laplacian in Bounded and in Unbounded Domains with Perforated Boundaries”, J. Differ. Equ., 216:2 (2005), 502–522
27. Chechkin G., Perez M., Yablokova E., “Non-Periodic Boundary Homogenization and “Light” Concentrated Masses”, Indiana Univ. Math. J., 54:2 (2005), 321–348
28. G. A. Chechkin, “Homogenization of solutions to problems for the Laplace operator in unbounded domains with many concentrated masses on the boundary”, Journal of Mathematical Sciences (New York), 139:1 (2006), 6351
29. 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
30. G. A. Chechkin, Yu. O. Koroleva, L.-E. Persson, “On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure”, J Inequal Appl, 2007 (2007), 1
31. M. N. Zubova, T. A. Shaposhnikova, “Homogenization of the Variational Inequality Corresponding to a Problem with Rapidly Varying Boundary Conditions”, Math. Notes, 82:4 (2007), 481–491
32. Chechkin G.A., Gadyl'Shin R.R., “On Boundary-Value Problems for the Laplacian in Bounded Domains with Micro Inhomogeneous Structure of the Boundaries”, Acta. Math. Sin.-English Ser., 23:2 (2007), 237–248
33. Mel'nik T.A., Vashchuk P.S., “Homogenization of a Boundary Value Problem with Mixed Type of Boundary Conditions in a Thick Junction”, Differ. Equ., 43:5 (2007), 696–703
34. D Borisov, G Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J Phys A Math Theor, 42:36 (2009), 365205
35. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition”, Ann Henri Poincaré, 2010
36. D. Borisov, R. Bunoiu, G. Cardone, “Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows”, J Math Sci, 2011
37. 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
38. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “On a waveguide with an infinite number of small windows”, Comptes Rendus Mathematique, 349:1-2 (2011), 53
39. Chechkina A.G., “On Singular Perturbations of a Steklov-Type Problem with Asymptotically Degenerate Spectrum”, Dokl. Math., 84:2 (2011), 695–698
40. G. A. Chechkin, Yu. O. Koroleva, L.-E. Persson, P. Wall, “On Spectrum of the Laplacian in a Circle Perforated along the Boundary: Application to a Friedrichs-Type Inequality”, International Journal of Differential Equations, 2011 (2011), 1
41. G.A. Chechkin, D. Cioranescu, A. Damlamian, A.L. Piatnitski, “On boundary value problem with singular inhomogeneity concentrated on the boundary”, Journal de Mathématiques Pures et Appliquées, 98:2 (2012), 115
42. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys, 2012
43. T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition”, Sb. Math., 205:10 (2014), 1492–1527
44. R. R. Gadyl'shin, S. V. Repjevskij, E. A. Shishkina, “On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 76–90
45. 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
46. T. F. Sharapov, “On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case”, Ufa Math. J., 8:2 (2016), 65–94
47. D. B. Davletov, D. V. Kozhevnikov, “The problem of Steklov type in a half-cylinder with a small cavity”, Ufa Math. J., 8:4 (2016), 62–87
48. Borisov D. Cardone G. Durante T., “Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. R. Soc. Edinb. Sect. A-Math., 146:6 (2016), 1115–1158
49. A. G. Chechkina, “Homogenization of spectral problems with singular perturbation of the Steklov condition”, Izv. Math., 81:1 (2017), 199–236
50. 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
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