Embeddings of Anisotropic Sobolev Spaces

  • D. E. Edmunds
  • R. M. Edmunds

Abstract

During the last twenty years a great deal of effort has been put into the analysis of embeddings of Sobolev spaces from the standpoint of approximation and entropy numbers. These numbers enable classifications of compact embedding maps to be made, and their intimate connection with eigenvalues of elliptic operators helps to explain the emphasis placed upon them by the strong group in the Soviet Union centred around Birman and Solomjak. Estimates have been obtained for the embeddings of the Sobolev space W k,p (Ω) in suitable Lebesgue spaces L q (Ω) when Ω is a bounded open subset of ℝ n , and even for certain types of unbounded sets Ω. When kp = n, estimates for these numbers are known for the compact embedding of W k,p (Ω) in particular Orlicz spaces.

Keywords

Entropy Expense Hull Stein Bedding 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R. A.: Sobolev spaces. New York, London: Academic Press 1975.Google Scholar
  2. 2.
    Besov, O. V., V. P. Il’ in, & S. N. Nlkol’slcn: Integral representations of functions and imbedding theorems, I and II. Washington, New York, Toronto, London, Sydney: Winston/Wiley 1978 and 1979.Google Scholar
  3. 3.
    Birman, M. S., & M. Z. Solomjak: Quantitative analysis in Sobolev embedding theorems and applications to spectral theory, Amer. Math. Soc. Translations114 (1980).Google Scholar
  4. 4.
    Borzov, V. V.: On some applications of piecewise polynomial approximations of functions of anisotropic classes Wpr, Soviet Math. Dokl. 12 (1971), 804–807.Google Scholar
  5. 5.
    Borzov, V. V.: Some applications of theorems on piecewise polynomial approximations of functions in anisotropic classes Wpr, Problemy Mat. Fiz., vyp. 6, Izdat. Leningrad Univ., Leningrad (1973), 53–67 (Russian).Google Scholar
  6. 6.
    Carl, B.: Entropy numbers, s-numbers and eigenvalues problem, J. Functional Analysis41 (1981), 290–306.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Carl, B., & H. Triebel: Inequalities between eigenvalues, entropy numbers and related quantities in Banach spaces, Math. Ann. 251 (1980), 129–133.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Edmunds, D. E.: Embeddings of Sobolev spaces, Proc. Spring School `Nonlinear analysis, function spaces and applications’, Teubner-Texte Math. 19 (Teubner, Leipzig 1979), 38–58.Google Scholar
  9. 9.
    Edmunds, D. E., & R. M. Edmunds: Entropy and approximation numbers of embeddings in Orlicz spaces, J. Lond. Math. Soc. 32 (1985), 528–538.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Edmunds, D. E., & V. B. Moscatelli: Fourier approximation and embeddings of Sobolev spaces, Dissertationes Mathematicae145 (1977), 1–50.MathSciNetGoogle Scholar
  11. 11.
    Fučik, S., O. John, & A. Kufner: Function spaces. Prague: Academia 1977.MATHGoogle Scholar
  12. 12.
    König, H.: A formula for the eigenvalues of a compact operator, Studia Math. 65 (1979), 141–146.MATHMathSciNetGoogle Scholar
  13. 13.
    König, H.: Some inequalities for the eigenvalues of a compact operator, to appear.Google Scholar
  14. 14.
    Pietsch, A.: Operator ideals. Berlin: VEB Deutscher Verlag der Wissenschaften 1978.Google Scholar
  15. 15.
    Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton: Princeton University Press 1970.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • D. E. Edmunds
    • 1
    • 2
  • R. M. Edmunds
    • 1
    • 2
  1. 1.Mathematics DivisionUniversity of SussexUK
  2. 2.Department of Pure MathematicsUniversity CollegeCardiffUK

Personalised recommendations