Advertisement

Contributions of V.G. Maz’ya to analysis of singularly perturbed boundary value problems

  • A. B. Movchan
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 109)

Abstract

It is a great pleasure and honour for me to contribute this article to the volume published on the occasion of the 60th birthday of Prof. V.G. Maz’ya. The objective of this paper is to give a review of contributions of Maz’ya and his colleagues to the development and justification of the method of compound asymptotic expansions. This work created a broad and powerful asymptotic theory, which opened new perspectives in the study of fields in singularly perturbed domains.

Keywords

Asymptotic Expansion Model Problem Boundary Integral Equation Remainder Function Boundary Layer Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    ClArlet, P.G.: Plates and junctions in elastic multi-structures, Masson, Paris, 1990.Google Scholar
  2. [2]
    ClArlet, P.G.: Mathematical elasticity. Volume II: theory of plates, Elsevier, 1997.Google Scholar
  3. [3]
    ClArlet, P.G. and Destuynder, P.: A justification of the two-dimensional plate model, J. Mécanique 18 (1979), 315–344.Google Scholar
  4. [4]
    Cole, J.D.: Perturbation methods in applied mathematics, Blaisdell, Waltham, 1968.zbMATHGoogle Scholar
  5. [5]
    Il’in, A.M.: Boundary value problem for a second order elliptic differential equation in a domain with a slit, I. Two-dimensional case, Matem. Sbornik 99 (1976), No 4, 514–537 (in Russian).Google Scholar
  6. [6]
    Il’in, A.M.: Boundary value problem for a second order elliptic differential equation in a domain with a thin slit, II. Domain with a small hole, Matern. Sbornik 103 (1977), No 2, 265–284 (in Russian).Google Scholar
  7. [7]
    Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, 102, American Mathematical Society, Providence, 1992.Google Scholar
  8. [8]
    Kozlov, V.A., Movchan, A.B. and Maz’ya, V.G.: Asymptotic analysis of a mixed boundary value problem in a multi-structure, Asymptotic analysis 8 (1994), 105–143.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Kozlov, V.A., Movchan, A.B. and Maz’ya, V.G.: Asymptotic representation of elastic fields in a multi-structure, Asymptotic analysis 11 (1995), 343–415.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Kozlov, V.A., Movchan, A.B. and Maz’ya, V.G.: Fields in non-degenerate 1D-3D elastic multi-structures, LiTH-MAT-R-96-14, 1996, Preprint, Linköping University.Google Scholar
  11. [11]
    Kozlov, V.A., Maz’ya, V.G. and Movchan, A.B.: Asymptotic analysis of fields in multi-structures, Oxford University Press (to appear).Google Scholar
  12. [12]
    Maz’ya, V.G. and HÄnler, M.: Approximation of solutions to the Neumann problem in disintegrating domains. Math. Nachr. 162 (1993), 261–278.MathSciNetGoogle Scholar
  13. [13]
    Maz’ya, V.G. and Mahnke, R.: Asymptotics of the solution of a boundary integral equation under a small perturbation of a corner, Zeitschrift für Analysis und ihre Anwendungen 11 (1992), No 2, 173–182.MathSciNetGoogle Scholar
  14. [14]
    Maz’ya, V.G. and Nazarov, S.A.: The limit passage paradox in solutions of boundary value problems in approximation of smooth domains by polygons, Izv. Acad. Nauk SSSR 50 (1986), No 6, 1156–1177 (in Russian).MathSciNetGoogle Scholar
  15. [15]
    Maz’ya, V.G. and Nazarov, S.A.: On the singularities of solutions of the Neumann problem at a conical point. Siberian Mathem. Journal30 (1989), No 3, 52–63 (in Russian).MathSciNetGoogle Scholar
  16. [16]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: On asymptotics of solutions to boundary value problems with variation of a domain near conical points. Doklady Acad. Nauk SSSR 249 (1979), No 1, 94–96 (in Russian).Google Scholar
  17. [17]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: On asymptotics of solutions of elliptic boundary value problems in singularly perturbed domains. Problemy Matern. Analiza 8 (1981), 72–153 (in Russian).Google Scholar
  18. [18]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: On asymptotics of solutions of the Dirichlet problem in a three-dimensional domain with a thin body excluded. Doklady Acad. Nauk SSSR 256 (1981), No 1, 37–39 (in Russian).Google Scholar
  19. [19]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: The absence of the De Giorgi theorem for elliptic equations with complex coefficients. Zap. Nauchn. Seminar. LOMI 115 (1982), 156–168 (in Russian).Google Scholar
  20. [20]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone. Matem. Sbornik 122 (1983), No 4, 435–456 (in Russian).Google Scholar
  21. [21]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: Evaluation of the asymptotic form of the intensity coefficients on approaching corner or conical points. U.S.S.R. Comput.Maths.Math.Phys. 23 (1983), No 2, 50–58.CrossRefGoogle Scholar
  22. [22]
    Maz’ya, V.G., Nazarov, S.A. and Plamenewskii, B.A.: Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small holes. Mathematics Izvestija 48 (1984), No 2, 347–371 (in Russian).Google Scholar
  23. [23]
    Mazja, W.G., Nazarow, S.A. and Plamenewskii, B.A.: Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. Akademie-Verlag, Berlin, B.l, 1991; B.2, 1992.Google Scholar
  24. [24]
    Maz’ya, V.G. and Slutskii, A.S.: Homogenization of a differential operator on a fine periodic curvilinear mesh, Math. Nachr. 133 (1986), 107–133.MathSciNetGoogle Scholar
  25. [25]
    Maz’ya, V.G. and Slutskii, A.S.: Homogenization of difference equations with rapidly oscillating coefficients, Seminar Analysis 1986/87, Akad. Wiss. DDR. Inst. Math. 63–92, Berlin, 1987.Google Scholar
  26. [26]
    Nayfeh, A.H.: Perturbation methods, New York, Chichester: Wiley, 1973.zbMATHGoogle Scholar
  27. [27]
    Van Dyke, M.D.: Perturbation methods in fluid mechanics, Academic Press, New York, 1964.zbMATHGoogle Scholar
  28. [28]
    Vishik, M.I. and Lyusternik, L.A.: Regular perturbation and boundary layer for differential equations with a small parameter. Uspekhi Matem. Nauk 15 (1957), No 3, 3–80 (in Russian).MathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • A. B. Movchan
    • 1
  1. 1.Department of Math. SciencesUniversity of LiverpoolLiverpoolUK

Personalised recommendations