Advertisement

Ukrainian Mathematical Journal

, Volume 19, Issue 5, pp 509–530 | Cite as

A theorem on homeomorphisms and the Green's function for general elliptic boundary problems

  • Yu. M. Berezanskii
  • Ya. A. Roitberg
Article

Keywords

Boundary Problem Elliptic Boundary Elliptic Boundary Problem General Elliptic Boundary 
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.

Literature cited

  1. 1.
    J. L. Lions and E. Magenes, Problemi ai Limiti non Omogenei. (III), Ann. Scuola Norm. Super., Pisa, Ser. III,15, Nos. 1–2, 39–101 (1961).Google Scholar
  2. 2.
    J. L. Lions and E. Magenes, Problemi ai Limiti non Omogenei. (V), Ann. Scuola Norm. Super., Pisa, Ser. III,16., No. 1, 1–44 (1962).Google Scholar
  3. 3.
    E. Magenes, Spazi di Interpolatione ed Equazioni a Derivate Parziali, Conferenza tenuta al VII Congresso dell' U.M.I., Genoa (1963).Google Scholar
  4. 4.
    Yu. M. Berezanskii, S. G. Krein, and Ya. A. Roitberg, A Theorem on Homeomorphisms and the Local Increase in Smoothness up to the Boundary of Solutions of Elliptic Equations, Dokl. Akad. Nauk SSSR,118, No. 4, 745–748 (1963).Google Scholar
  5. 5.
    Ya. A. Roitberg, Elliptic Problems with Non-Homogeneous Boundary Conditions and the Local Increase in the Smoothness of Generalized Solutions up to the Boundary, Dokl. Akad. Nauk SSSR,157, No. 4, 798–801 (1964).Google Scholar
  6. 6.
    Ya. A. Roitberg, A Theorem on Homeomorphisms Generated by Elliptic Operators and the Local Increase in the Smoothness of Generalized Solutions, Ukrain. Mat. Zh.,17, No. 5, 122–129 (1965).Google Scholar
  7. 7.
    Yu. M. Berezanskii, Eigenfunction Expansions of Self-Adjoint Operators [in Russian], Izdat. Naukova Dumka, Kiev (1965).Google Scholar
  8. 8.
    J. Odhnoff, Operators Generated by Differential Problems with Value Parameter in Equation and Boundary Condition, Lund (1959).Google Scholar
  9. 9.
    J. Ercolano and M. Schechter, Spectral Theory for Operators Generated by Elliptic Boundary Problems, Comm. Pure Appl. Math.,18, Nos. 1–2, 83–105 (1965).Google Scholar
  10. 10.
    J. Ercolano and M. Schechter, Spectral Theory for Operators Generated by Elliptic Boundary Problems, Comm. Pure. App. Math.,18, No. 3, 397–414 (1965).Google Scholar
  11. 11.
    V. V. Barkovskii and Ya. A. Roitberg, On the Minimal and Maximal Operators Associated with a General Elliptic Problem with Non-Homogeneous Boundary Conditions, Ukrain. Mat. Zh.,18, No. 2, 91–97 (1966).Google Scholar
  12. 12.
    V. V. Barkovskii, On the Green's Function of a Self-Adjoint Elliptic Operator Generated by a Differential Expression and by Non-Homogeneous Boundary Conditions, Ukrain. Mat. Zh.,18, No. 2, 84–91 (1966).Google Scholar
  13. 13.
    V. V. Barkovskii, An Eigenfunction Expansion of Self-Adjoint Operators Corresponding to General Elliptic Problems with an Eigenvalue in the Boundary Conditions, Ukrain. Mat. Zh.,19, No. 1 (1967).Google Scholar
  14. 14.
    M. I. Vishik and S. L. Sobolev, The General Statement of Certain Boundary Value Problems for Elliptic Partial Differential Equations, Dokl. Akad. Nauk SSSR,111, No. 3, 521–523 (1956).Google Scholar
  15. 15.
    Yu. M. Berezanskii, On the Smoothness up to the Boundary of the Domain of the Spectral Function of a Self-Adjoint Elliptic Differential Operator, Dokl. Akad. Nauk SSSR,152, No. 3, 511–514 (1963).Google Scholar
  16. 16.
    S. G. Krein and A. S. Simonov, A Theorem on Homeomorphisms and Quasi-Linear Equations, Dokl. Akad. Nauk SSSR,167, No. 6, 1226–1229 (1966).Google Scholar
  17. 17.
    Ya. A. Roitberg, The Solvability of General Boundary Problems for Elliptic Equations when there are Power Singularities on the Right Sides, Ukrain, Mat. Zh.,20, No. 1 (1968).Google Scholar
  18. 18.
    A. Ya. Povzner, The Expansion of Arbitrary Functions in Terms of the Eigenfunctions of the Operator — Δu + cu, Mat. Sb.,32, No.1, 109–156 (1953).Google Scholar
  19. 19.
    F. E. Browder, The Eigenfunction Expansion Theorem for the General Self-Adjoint Singular Elliptic Partial Differential Operator, Proc. Nat. Acad. Sci. USA,40, No. 6, 454–467 (1954).Google Scholar
  20. 20.
    F. E. Browder, On the Spectral Theory of Elliptic Differential Operators, I, Math. Ann.,142, 22–130 (1961).Google Scholar
  21. 21.
    K. Moren, Hilbert Space Methods [in Russian], Izdat. Mir, Moscow (1965).Google Scholar
  22. 22.
    V, A. Il'in and I. A. Shishmarev, On the Equivalence of Systems of Generalized and Classical Eigenfunctions, Izv. Akad. Nauk SSSR, Ser. Matem.,24, No. 5, 757–774 (1960).Google Scholar
  23. 23.
    Yu. M. Berezanskii and Ya. A. Roitberg, On the Smoothness up to the Boundary of a Domain of the Resolvent Kernel of an Elliptic Operator, Ukrain. Mat. Zh.,15, No. 2, 185–189 (1963).Google Scholar
  24. 24.
    Yu. P. Krasovskii, An Investigation of the Green's Function, Uspekhi Mat. Nauk,20, No. 5, 267–268 (1965).Google Scholar
  25. 25.
    M. Sh. Birman, Perturbations of the Continuous Spectrum of a Singular Elliptic Operator Under a Variation of the Boundary and of the Boundary Conditions, Vestnik Leningr. Cos. Univ., Ser. Matem., Mekh., Astr., No. 1, 22–55 (1962).Google Scholar
  26. 26.
    M. Sh. Birman and Z. S. Solomyak, On the Approximation of Functions from the Wα p classes byPiecewise-Polynomial Functions, Dokl. Akad. Nauk SSSR,171, No. 5, 1015–1018 (1966).Google Scholar
  27. 27.
    M. Schechter, General Boundary Value Problems for Elliptic Partial Differential Equations, Comm. Pure Appl. Math.,12, No. 3, 457–486 (1959).Google Scholar
  28. 28.
    N. Bourbaki, Espaces Vectoriels Topologiques, Herman, Paris (1953).Google Scholar
  29. 29.
    S. G. Krein, Address to the All-Union Conference on Functional Analysis and Its Applications, Baku (1959).Google Scholar
  30. 30.
    S. G. Krein, On an Interpolation Theorem in Operator Theory, Dokl. Akad. Nauk SSSR,130, No. 3, 491–494 (1960).Google Scholar
  31. 31.
    J. L. Lions, Éspaces Intermédiaires Entre Éspaces Hilbertiens et Applications, Bull. Math. Soc. Math. Phys., RPR,2(50), No. 3, 419–432 (1959).Google Scholar
  32. 32.
    M. Schechter, On LP Estimates and Regularity. I, Amer. J. Math.,85, No. 1–13 (1963).Google Scholar
  33. 33.
    M. Schechter, On LP Estimates and Regularity. II, Math. Scand.,13, No. 1, 47–69 (1963).Google Scholar
  34. 34.
    M. S. Agranovich, Elliptic Singular Integro-Differential Operators, Uspekhi Mat. Nauk,20, No. 5, 3–120 (1965).Google Scholar
  35. 35.
    L. P. Volevich, The Solvability of Boundary Problems for General Elliptic Systems, Mat. Sb.,68, No. 3, 373–416 (1965).Google Scholar
  36. 36.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces [in Russian], Fizmatgiz, Moscow (1959).Google Scholar
  37. 37.
    M. Sh. Birman and M. Z. Solomyak, On the Estimates of Singularities of Integral Operators, I, Vestnik Leningr. Gos. Univ., Ser. Mat., Mekh., Astr., No. 7 (1967).Google Scholar

Copyright information

© Consultants Bureau 1969

Authors and Affiliations

  • Yu. M. Berezanskii
    • 1
  • Ya. A. Roitberg
    • 1
  1. 1.Chernigovskii Pedagogical Institute of MathematicsAcademy of Sciences of the UkrSSRKiev

Personalised recommendations