Open Access

Elliptic boundary-value problems in Hörmander spaces


Abstract

We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.

Key words: Elliptic problem, Hörmander space, extended Sobolev scale, RO-varying function, Fredholm property, a priori estimate, local regularity


Full Text






Article Information

TitleElliptic boundary-value problems in Hörmander spaces
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 4, 295-310
MathSciNet   MR3591082
zbMATH 06742113
Milestones  Received 19/09/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

Anna Anop
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine; Chernihiv National Pedagogical University, 53 Het’mana Polubotka, Chernihiv, 14013, Ukraine

Tetiana Kasirenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine


Export article

Save to Mendeley



Citation Example

Anna Anop and Tetiana Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology 22 (2016), no. 4, 295-310.


BibTex

@article {MFAT911,
    AUTHOR = {Anop, Anna and Kasirenko, Tetiana},
     TITLE = {Elliptic boundary-value problems in Hörmander spaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {4},
     PAGES = {295-310},
      ISSN = {1029-3531},
  MRNUMBER = {MR3591082},
 ZBLNUMBER = {06742113},
       URL = {http://mfat.imath.kiev.ua/article/?id=911},
}


References

  1. M. S. Agranovich, Elliptic boundary problems, Partial differential equations, IX, Encyclopaedia Math. Sci., vol. 79, Springer, Berlin, 1997,  MathSciNet CrossRef
  2. A. V. Anop, Elliptic boundary-value problems in a multiply connected domain on the extended Sobolev scale, Zb. Pr. Inst. Mat. Nats. Akad. Nauk Ukr., vol. 10, no. 2, 2013, pp. 37-59 (Ukrainian).
  3. A. V. Anop, Elliptic boundary-value problems for systems of differential equations in the spaces of generalized smoothness, Zb. Pr. Inst. Mat. Nats. Akad. Nauk Ukr., vol. 11, no. 2, 2014, pp. 7-34 (Ukrainian).
  4. A. V. Anop, A general elliptic boundary-value problem on the extended Sobolev scale, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2014), no. 4, 7-14 (Ukrainian).
  5. A. V. Anop and A. A. Murach, Parameter-elliptic problems and interpolation with a function parameter, Methods Funct. Anal. Topology 20 (2014), no. 2, 103-116.  MathSciNet
  6. A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J. 66 (2014), no. 7, 969-985.  MathSciNet CrossRef
  7. V. G. Avakumovi, O jednom O-inverznom stavu, Rad Jugoslovenske Akad. Znatn. Umjetnosti 254 (1936), 167-186.
  8. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
  9. Joran Bergh and Jorgen Lofstrom, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976.  MathSciNet
  10. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1989.  MathSciNet
  11. Yu. V. Egorov, Linear differential equations of principal type, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.  MathSciNet
  12. C. Foias and J.-L. Lions, Sur certains theor\`emes dinterpolation, Acta Sci. Math. Szeged 22 (1961), 269-282.  MathSciNet
  13. Lars Hormander, Linear partial differential operators, Grundlehren Math. Wiss 116, Springer-Verlag, Berlin, 1963.  MathSciNet
  14. Lars Hormander, The analysis of linear partial differential operators. II. Differential operators with constant coefficients, Grundlehren Math. Wiss 257, Springer-Verlag, Berlin, 1983.  MathSciNet CrossRef
  15. Lars Hormander, The analysis of linear partial differential operators. III, Grundlehren Math. Wiss 274, Springer-Verlag, Berlin, 1985.  MathSciNet
  16. N. Jacob, Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups, Imperial College Press, London, 2001.  MathSciNet CrossRef
  17. V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs, vol. 52, American Mathematical Society, Providence, RI, 1997.  MathSciNet
  18. J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren Math. Wiss 181, Springer-Verlag, New York-Heidelberg, 1972.  MathSciNet
  19. W. Matuszewska, On a generalization of regularly increasing functions, Studia Math. 24 (1964), 271-279.  MathSciNet
  20. V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian Math. J. 57 (2005), no. 5, 817-825.  MathSciNet CrossRef
  21. V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary value problems. II, Ukrainian Math. J. 58 (2006), no. 3, 398-417.  MathSciNet CrossRef
  22. V. A. Mikhailets and A. A. Murach, A regular elliptic boundary value problem for a homogeneous equation in a two-sided refined scale of spaces, Ukrainian Math. J. 58 (2006), no. 11, 1748-1767.  MathSciNet CrossRef
  23. V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary value problems. III, Ukrainian Math. J. 59 (2007), no. 5, 744-765.  MathSciNet CrossRef
  24. V. A. Mikhailets and A. A. Murach, An elliptic boundary value problem in a two-sided refined scale of spaces, Ukrainian Math. J. 60 (2008), no. 4, 574-597.  MathSciNet CrossRef
  25. V. A. Mikhailets and A. A. Murach, Elliptic operators on a closed compact manifold, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2009), no. 3, 29-35 (Russian).  MathSciNet
  26. V. A. Mikhailets and A. A. Murach, The refined Sobolev scale, interpolation, and elliptic problems, Banach J. Math. Anal. 6 (2012), no. 2, 211-281.  MathSciNet CrossRef
  27. V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J. 65 (2013), no. 3, 435-447.  MathSciNet CrossRef
  28. V. A. Mikhailets and A. A. Murach, Hormander spaces, interpolation, and elliptic problems, De Gruyter Studies in Mathematics, vol. 60, De Gruyter, Berlin, 2014.  MathSciNet CrossRef
  29. V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math. 67 (2015), no. 1-2, 135-152.  MathSciNet CrossRef
  30. Aleksandr A. Murach and Tetiana Zinchenko, Parameter-elliptic operators on the extended Sobolev scale, Methods Funct. Anal. Topology 19 (2013), no. 1, 29-39.  MathSciNet
  31. Fabio Nicola and Luigi Rodino, Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators. Theory and Applications, vol. 4, Birkhauser Verlag, Basel, 2010.  MathSciNet CrossRef
  32. V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), no. 2, 349-515.  MathSciNet
  33. Boris P. Paneah, The oblique derivative problem, Mathematical Topics, vol. 17, Wiley-VCH, Berlin, 2000.  MathSciNet
  34. J. Peetre, On interpolation functions, Acta Sci. Math. (Szeged) 27 (1966), 167-171.  MathSciNet
  35. J. Peetre, On interpolation functions. II, Acta Sci. MAth. (Szeged) 29 (1968), 91-92.  MathSciNet
  36. Yakov Roitberg, Elliptic boundary value problems in the spaces of distributions, Mathematics and its Applications, vol. 384, Kluwer Academic Publishers Group, Dordrecht, 1996.  MathSciNet CrossRef
  37. Yakov Roitberg, Boundary value problems in the spaces of distributions, Mathematics and its Applications, vol. 498, Kluwer Academic Publishers, Dordrecht, 1999.  MathSciNet CrossRef
  38. Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976.  MathSciNet
  39. G. Slenzak, Elliptic problems in a refined scale of spaces, Moscow Univ. Math. Bull. 29 (1974), no. 3--4, 80-88.
  40. Hans Triebel, Interpolation theory, function spaces, differential operators, Johann Ambrosius Barth, Heidelberg, 1995.  MathSciNet
  41. Hans Triebel, The structure of functions, Monographs in Mathematics, vol. 97, Birkhauser Verlag, Basel, 2001.  MathSciNet CrossRef
  42. L. R. Volevich and B. P. Panejah, Some spaces of generalized functions and embedding theorems, Russian Math. Surveys 20 (1965), no. 1, 1-73.
  43. T. N. Zinchenko and A. A. Murach, Douglis-Nirenberg elliptic systems in Hormander spaces, Ukrainian Math. J. 64 (2013), no. 11, 1672-1687.  MathSciNet CrossRef
  44. Tetiana N. Zinchenko and Aleksandr A. Murach, Petrovskii elliptic systems in the extended Sobolev scale, J. Math. Sci. (N. Y.) 196 (2014), no. 5, 721-732.  MathSciNet CrossRef


All Issues