General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mat. Sb.:

Personal entry:
Save password
Forgotten password?

Mat. Sb. (N.S.), 1978, Volume 106(148), Number 4(8), Pages 604–621 (Mi msb2609)  

This article is cited in 55 scientific papers (total in 55 papers)

The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain

E. Ya. Khruslov

Abstract: The second boundary value problem is considered for the equation $\Delta u-cu=f$ in a domain $G^{(s)}$ of complicated structure of the form $G^{(s)}=\mathbf R_n\setminus F^{(s)}$, where $F^{(s)}$ is a closed finely partitioned set lying in a domain $\Omega\subset\mathbf R_n$ ($n\geqslant 2$) for all $s=1,2,…$. The asymptotic behavior of a solution $u^{(s)}(x)$ of this problem is studied as $s\to\infty$, when $F^{(s)}$ becomes more and more finely divided and is situated in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero. It is proved that under specific conditions $u^{(s)}(x)$ converges in $\mathbf R_n\setminus\overline\Omega$ to a function $u(x)$ that is a solution of a conjugation problem. Sufficient conditions for convergence are formulated.
Bibliography: 9 titles.

Full text: PDF file (1508 kB)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Sbornik, 1979, 35:2, 266–282

Bibliographic databases:

UDC: 517.946.9
MSC: 35G15, 35B40
Received: 21.11.1977

Citation: E. Ya. Khruslov, “The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain”, Mat. Sb. (N.S.), 106(148):4(8) (1978), 604–621; Math. USSR-Sb., 35:2 (1979), 266–282

Citation in format AMSBIB
\by E.~Ya.~Khruslov
\paper The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain
\jour Mat. Sb. (N.S.)
\yr 1978
\vol 106(148)
\issue 4(8)
\pages 604--621
\jour Math. USSR-Sb.
\yr 1979
\vol 35
\issue 2
\pages 266--282

Linking options:

    SHARE: FaceBook Twitter Livejournal

    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. V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Hà Tiên Ngoan, “Averaging and $G$-convergence of differential operators”, Russian Math. Surveys, 34:5 (1979), 69–147  mathnet  crossref  mathscinet  zmath
    2. Fenchenko V., Khruslov E., “Asymptotic Solutions of Differential-Equations with Fast Oscillating and Degenerating Coefficient Matrix”, no. 4, 1980, 25–29  mathscinet  zmath  isi
    3. Fenchenko V., Khruslov E., “Asymptotics of Differential-Equation Solutions with a Largely Oscillating Coefficient Matrix Not Satisfying the Condition of Uniform Limitation”, no. 4, 1981, 23–27  mathscinet  zmath  isi
    4. Berliand L., Chudinovich I., “Homogenization of Boundary-Value-Problems for Higher-Order Differential-Operators in Domains with Holes”, 272, no. 4, 1983, 777–780  mathscinet  isi
    5. Berlyand L., “Vibrations of an Elastic Body with a Large Number of Small Emptinesses”, no. 2, 1983, 3–5  mathscinet  zmath  isi
    6. Berlyand L., “Asymptotic Description of the Thin Plate with a Large Number of Small Holes”, no. 10, 1983, 5–8  mathscinet  zmath  isi
    7. Kovalevsky A., “Averaging of Varying Variation Problems”, no. 8, 1988, 6–8  mathscinet  isi
    8. Valeria Chiadò Piat, “Convergence of minima for nonequicoercive functionals and related problems”, Annali di Matematica, 157:1 (1990), 251  crossref  mathscinet  zmath  isi
    9. V. V. Zhikov, “Asymptotic problems connected with the heat equation in perforated domains”, Math. USSR-Sb., 71:1 (1992), 125–147  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. Satoshi Kaizu, “Behavior of solutions of the Poisson equation under fragmentation of the boundary of the domain”, Japan J Appl Math, 7:1 (1990), 77  crossref
    11. Valeria Chiadòpiat, Anneliese Defranceschi, “Asymptotic behaviour of quasi-linear problems with Neumann boundary conditions on perforated domains”, Applicable Analysis, 36:1-2 (1990), 65  crossref
    12. Zhikov V., “Problems of Function Extensions Related to the Theory of Homogenization”, Differ. Equ., 26:1 (1990), 34–44  mathnet  mathscinet  zmath  isi
    13. Satoshi Kaizu, “The Poisson Equation with Nonautonomous Semilinear Boundary Conditions in Domains with Many Time Holes”, SIAM J Math Anal, 22:5 (1991), 1222  crossref  mathscinet  zmath  isi
    14. V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459  mathnet  crossref  mathscinet  zmath  isi
    15. E. Acerbi, V. ChiadòPiat, G. Dal Maso, D. Percivale, “An extension theorem from connected sets, and homogenization in general periodic domains”, Nonlinear Analysis: Theory, Methods & Applications, 18:5 (1992), 481  crossref
    16. A. A. Kovalevsky, “$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 431–460  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    17. Andrea Braides, Valeria Chiadò Piat, “Remarks on the homogenization of connected media”, Nonlinear Analysis: Theory, Methods & Applications, 22:4 (1994), 391  crossref
    18. Kovalevskii A., “Uniform Boundedness of Solutions to Nonlinear Elliptic Variational-Inequalities in Variable Domains”, Differ. Equ., 30:8 (1994), 1270–1273  mathnet  mathscinet  isi
    19. Cortesani Guido, “Asymptotic behaviour of a sequence of Neumann problems”, Communications in Partial Differential Equations, 22:9-10 (1997), 1691  crossref
    20. Satoshi Kaizu, “Homogenization of eigenvalue problems for the Laplace operator with nonlinear terms in domains in many tiny holes”, Nonlinear Analysis: Theory, Methods & Applications, 28:2 (1997), 377  crossref
    21. Briane M., “Homogenization of the Torsion Problem and the Neumann Problem in Nonregular Periodically Perforated Domains”, Math. Models Meth. Appl. Sci., 7:6 (1997), 847–870  crossref  mathscinet  zmath  isi
    22. L. Berlyand, E. Khruslov, “Homogenization of Harmonic Maps and Superconducting Composites”, SIAM J Appl Math, 59:5 (1999), 1892  crossref  mathscinet  zmath  isi
    23. Christopher L. Holloway, Edward F. Kuester, “Equivalent boundary conditions for a perfectly conducting periodic surface with a cover layer”, Radio Sci, 35:3 (2000), 661  crossref
    24. Briane M., “Increase of Dimension by Homogenization”, Potential Anal., 14:3 (2001), 233–268  crossref  mathscinet  zmath  isi
    25. Damlamian, A, “Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?”, ESAIM-Control Optimisation and Calculus of Variations, 8 (2002), 555  crossref  isi
    26. Briane, M, “Homogenization of a class of non-uniformly elliptic monotonic operators”, Nonlinear Analysis-Theory Methods & Applications, 48:1 (2002), 137  crossref  isi  elib
    27. Felipe Alvarez, Jean-Philippe Mandallena, “Homogenization of multiparameter integrals”, Nonlinear Analysis: Theory, Methods & Applications, 50:6 (2002), 839  crossref
    28. Pankratov L., Piatnitski A., “Nonlinear “Double Porosity” Type Model”, C. R. Math., 334:5 (2002), 435–440  crossref  mathscinet  zmath  isi
    29. Berlyand L., Khruslov E., “Competition Between the Surface and the Boundary Layer Energies in a Ginzburg-Landau Model of a Liquid Crystal Composite”, Asymptotic Anal., 29:3-4 (2002), 185–219  mathscinet  zmath  isi
    30. L. S. Pankratov, V. A. Rybalko, “Asymptotic analysis of a double porosity model with thin fissures”, Sb. Math., 194:1 (2003), 123–150  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    31. Novikov A., “Modulational Stability of Cellular Flows”, Nonlinearity, 16:5 (2003), 1607–1639  crossref  mathscinet  zmath  adsnasa  isi
    32. B. Amaziane, M. Goncharenko, L. Pankratov, “ΓD-convergence for a class of quasilinear elliptic equations in thin structures”, Math Meth Appl Sci, 28:15 (2005), 1847  crossref  mathscinet  zmath  isi  elib
    33. Proc. Steklov Inst. Math., 250 (2005), 245–253  mathnet  mathscinet  zmath
    34. Berlyand L., Cioranescu D., Golovaty D., “Homogenization of a Ginzburg-Landau Functional”, C. R. Math., 340:1 (2005), 87–92  crossref  mathscinet  zmath  isi
    35. Berlyand L., Cioranescu D., Golovaty D., “Homogenization of a Ginzburg-Landau Model for a Nematic Liquid Crystal with Inclusions”, J. Math. Pures Appl., 84:1 (2005), 97–136  crossref  mathscinet  zmath  isi
    36. Amaziane, B, “Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media”, Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 136 (2006), 1131  crossref  isi  elib
    37. B. Amaziane, L. Pankratov, “On the homogenization of some linear problems in domains weakly connected by a system of traps”, Math Meth Appl Sci, 30:15 (2007), 1855  crossref  mathscinet  zmath  isi  elib
    38. Graeme W Milton, “New metamaterials with macroscopic behavior outside that of continuum elastodynamics”, New J Phys, 9:10 (2007), 359  crossref  isi
    39. Braides, A, “Homogenization of non-linear variational problems with thin low-conducting layers”, Applied Mathematics and Optimization, 55:1 (2007), 1  crossref  isi  elib
    40. A. V. Khrabustovskyi, “Klein–Gordon equation as a result of wave equation averaging on the Riemannian manifold of complex microstructure”, Zhurn. matem. fiz., anal., geom., 3:2 (2007), 213–233  mathnet  mathscinet  zmath  elib
    41. S. Kesavan, T. Muthukumar, “Low-cost control problems on perforated and non-perforated domains”, Proc Math Sci, 118:1 (2008), 133  crossref  mathscinet  zmath
    42. Milton G.W., Seppecher P., “Realizable Response Matrices of Multi-Terminal Electrical, Acoustic and Elastodynamic Networks at a Given Frequency”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 464:2092 (2008), 967–986  crossref  mathscinet  zmath  adsnasa  isi
    43. Amaziane, B, “On the homogenization of some double-porosity models with periodic thin structures”, Applicable Analysis, 88:10–11 (2009), 1469  crossref  isi  elib
    44. Mel'nyk, TA, “Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain”, Ukrainian Mathematical Journal, 61:4 (2009), 592  crossref  isi
    45. Biroli M., “Gamma-CONVERGENCE FOR STRONGLY LOCAL Dirichlet FORMS IN OPEN SETS WITH HOLES”, Potential Theory and Stochastics in Albac: Aurel Cornea Memorial Volume, Conference Proceedings, 2009, 35–47  isi
    46. Rudakova O.A., “On Gamma-Convergence of Integral Functionals Defined on Various Weighted Sobolev Spaces”, Ukr. Math. J., 61:1 (2009), 121–139  crossref  mathscinet  zmath  isi
    47. Sango M., “Homogenization of the Neumann Problem for a Quasilinear Elliptic Equation in a Perforated Domain”, Networks and Heterogeneous Media, 5:2 (2010), 361–384  crossref  isi  elib
    48. F. Cagnetti, L. Scardia, “An extension theorem in SBV and an application to the homogenization of the Mumford–Shah functional in perforated domains”, Journal de Mathématiques Pures et Appliquées, 95:4 (2011), 349  crossref
    49. Kovalevsky A.A., “Obstacle Problems in Variable Domains”, Complex Var. Elliptic Equ., 56:12, SI (2011), 1071–1083  crossref  mathscinet  zmath  isi
    50. A. A. Kovalevsky, “On the convergence of solutions of variational problems with bilateral obstacles in variable domains”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 151–163  mathnet  crossref  mathscinet  isi  elib
    51. Kovalevsky A.A., “On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains”, Nonlinear Anal.-Theory Methods Appl., 147 (2016), 63–79  crossref  mathscinet  zmath  isi  scopus
    52. A. A. Kovalevskii, “Variatsionnye zadachi s odnostoronnimi potochechno funktsionalnymi ogranicheniyami v peremennykh oblastyakh”, Tr. IMM UrO RAN, 23, no. 2, 2017, 133–150  mathnet  crossref  elib
    53. Alexander A. Kovalevsky, “Convergence of solutions of bilateral problems in variable domains and related questions”, Ural Math. J., 3:2 (2017), 51–66  mathnet  crossref
    54. A. A. Kovalevsky, “On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains”, Funct. Anal. Appl., 52:2 (2018), 147–150  mathnet  crossref  crossref  isi  elib
    55. A. A. Kovalevskii, “O skhodimosti reshenii variatsionnykh zadach s neyavnymi ogranicheniyami, zadannymi bystro ostsilliruyuschimi funktsiyami”, Tr. IMM UrO RAN, 24, no. 2, 2018, 107–122  mathnet  crossref  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
    Number of views:
    This page:344
    Full text:100

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019