Skip to main content
SpringerLink
Log in

Regular Elliptic Boundary-Value Problems in the Extended Sobolev Scale

  • Published:
Ukrainian Mathematical Journal Aims and scope

We investigate an arbitrary regular elliptic boundary-value problem given in a bounded Euclidean C ∞- domain. It is shown that the operator of the problem is bounded and Fredholm in appropriate pairs of Hörmander inner-product spaces. They are parametrized with the help of an arbitrary radial function RO-varying at ∞ and form the extended Sobolev scale. We establish a priori estimates for the solutions of the problem and study their local regularity on this scale. New sufficient conditions for the generalized partial derivatives of the solutions to be continuous are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Reinhold, Princeton (1965).

    MATH  Google Scholar 

  2. Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).

    Google Scholar 

  3. L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).

    Book  MATH  Google Scholar 

  4. L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. 3. Pseudo-Differential Operators, Springer, Berlin (1985).

    Google Scholar 

  5. J.-L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Dunod, Paris (1968).

    MATH  Google Scholar 

  6. Ya. A. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht (1996).

    Book  MATH  Google Scholar 

  7. H. Triebel, Theory of Function Spaces, Akademische Verlagsgeselleschaft Geest & Portig K.-G., Leipzig (1983).

    Book  Google Scholar 

  8. M. S. Agranovich, “Elliptic boundary problems,” in: Encyclopedia of Mathematical Sciences, Vol. 79, Partial Differential Equations, IX, Springer, Berlin (1997), pp. 1–144.

    Google Scholar 

  9. L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. 2. Differential Operators with Constant Coefficients, Springer, Berlin (1983).

    Google Scholar 

  10. N. Jacob, Pseudodifferential Operators and Markov Processes, Vols. 1–3, Imperial College Press, London (2001, 2002, 2005).

  11. F. Nicola and L. Rodino, Global Pseudodifferential Calculus on Euclidean Spaces, Birkhäuser, Basel (2010).

    Book  Google Scholar 

  12. B. Paneah, The Oblique Derivative Problem. The Poincaré Problem, Wiley, Berlin (2000).

    MATH  Google Scholar 

  13. H. Triebel, The Structure of Functions, Birkhäuser, Basel (2001).

    Book  MATH  Google Scholar 

  14. V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).

    Book  MATH  Google Scholar 

  15. E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).

    Book  MATH  Google Scholar 

  16. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge Univ. Press, Cambridge (1989).

    MATH  Google Scholar 

  17. V. A. Mikhailets and A. A. Murach, “Elliptic operators in a refined scale of functional spaces,” Ukr. Math. J., 57, No. 5, 817–825 (2005).

    Article  MathSciNet  Google Scholar 

  18. V. A. Mikhailets and A. A. Murach, “Improved scale of spaces and elliptic boundary-value problems. II,” Ukr. Math. J., 58, No. 3, 398–417 (2006).

    Article  MathSciNet  Google Scholar 

  19. V. A. Mikhailets and A. A. Murach, “Refined scale of spaces and elliptic boundary-value problems. III,” Ukr. Math. J., 59, No. 5, 744–765 (2007).

    Article  MathSciNet  Google Scholar 

  20. V. A. Mikhailets and A. A. Murach, “An elliptic operator with homogeneous regular boundary conditions in a two-sided refined scale of spaces,” Ukr. Math. Bull., 3, No. 4, 529–560 (2006).

    MathSciNet  Google Scholar 

  21. V. A. Mikhailets and A. A. Murach, “Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces,” Ukr. Math. J., 58, No. 11, 1748–1767 (2006).

    Article  MathSciNet  Google Scholar 

  22. V. A. Mikhailets and A. A. Murach, “Elliptic boundary-value problem in a two-sided refined scale of spaces,” Ukr. Math. J., 60, No. 4, 574–597 (2008).

    Article  MathSciNet  Google Scholar 

  23. V. A. Mikhailets and A. A. Murach, “The refined Sobolev scale, interpolation, and elliptic problems,” Banach J. Math. Anal., 6, No. 2, 211–281 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  24. V. I. Ovchinnikov, “The methods of orbits in interpolation theory,” Math. Rep. Ser. 1, No. 2, 349–515 (1984).

    MathSciNet  Google Scholar 

  25. V. A. Mikhailets and A. A. Murach, “Extended Sobolev scale and elliptic operators,” Ukr. Math. J., 65, No. 3, 435–447 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  26. V. A. Mikhailets and A. A. Murach, “On elliptic operators on a closed manifold,” Dop. Nats. Akad. Nauk Ukr., No. 3, 13–19 (2009).

  27. A. A. Murach, “On elliptic systems in Hörmander spaces,” Ukr. Math. J., 61, No. 3, 467–477 (2009).

    Article  MathSciNet  Google Scholar 

  28. T. N. Zinchenko and A. A. Murach, “Douglis–Nirenberg elliptic systems in Hörmander spaces,” Ukr. Math. J., 64, No. 11, 1672–1687 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. A. Murach and T. Zinchenko, “Parameter-elliptic operators on the extended Sobolev scale,” Meth. Func. Anal. Topol., 19, No. 1, 29–39 (2013).

    MathSciNet  MATH  Google Scholar 

  30. T. N. Zinchenko, “Elliptic systems in the extended Sobolev scale,” Dop. Nats. Akad. Nauk Ukr., No. 3, 14–20 (2013).

  31. T. N. Zinchenko and A. A. Murach, “Petrovskii elliptic systems in the extended Sobolev scale,” J. Math. Sci. (N. Y.), 196, No. 5, 721–732 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  32. V. G. Avakumović, “O jednom O-inverznom stavu,” Rad. Jugoslovenske Akad. Znatn. Umjetnosti, 254, 167–186 (1936).

    Google Scholar 

  33. W. Matuszewska, “On a generalization of regularly increasing functions,” Stud. Math., 24, 271–279 (1964).

    MathSciNet  MATH  Google Scholar 

  34. L. R. Volevich and B. P. Paneyakh, “Some spaces of generalized functions and embedding theorems,” Usp. Mat. Nauk, 20, No. 1, 3–74 (1965).

    MATH  Google Scholar 

  35. V. A. Mikhailets and A. A. Murach, Interpolation Hilbert Spaces between Sobolev Spaces, Preprint arXiv:1106.2049v2 [math.FA] (2012).

  36. G. Slenzak, “Elliptic problems in a refined scale of spaces,” Moscow Univ. Math. Bull., 29, No. 3–4, 80–88 (1974).

    MathSciNet  MATH  Google Scholar 

  37. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983).

    Book  MATH  Google Scholar 

  38. C. Foias¸ and J.-L. Lions, “Sur certains théorèmes d’interpolation,” Acta Sci. Math. (Szeged), 22, No. 3–4, 269–282 (1961).

    MathSciNet  MATH  Google Scholar 

  39. Yu. A. Brudnyi and N. Ya. Krugljak, Interpolation Functors and Interpolation Spaces, North-Holland, Amsterdam (1991).

  40. V. A. Mikhailets and A. A. Murach, “Interpolation with a function parameter and refined scale of spaces,” Meth. Funct. Anal. Topol., 14, No. 1, 81–100 (2008).

    MathSciNet  MATH  Google Scholar 

  41. J. Peetre, “On interpolation functions. II,” Acta Sci. Math. (Szeged), 29, No. 1–2, 91–92 (1968).

    MathSciNet  MATH  Google Scholar 

  42. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin (1976).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    A. V. Anop & A. A. Murach

Authors
  1. A. V. Anop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Murach
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 7, pp. 867–883, July, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anop, A.V., Murach, A.A. Regular Elliptic Boundary-Value Problems in the Extended Sobolev Scale. Ukr Math J 66, 969–985 (2014). https://doi.org/10.1007/s11253-014-0988-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0988-6

Keywords

Search

Navigation