A semihomogeneous elliptic problem with additional unknown functions in boundary conditions

TitleA semihomogeneous elliptic problem with additional unknown functions in boundary conditions
Publication TypeJournal Article
Year of Publication2015
AuthorsChepurukhina, IS
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2015.07.020
Issue7
SectionMathematics
Pagination20-28
Date Published7/2015
LanguageUkrainian
Abstract

We investigate an elliptic boundary-value problem for a homogeneous differential equation, the problem containing additional unknown functions in the boundary conditions. We prove that the operator corresponding to this problem is bounded and Noetherian in appropriate pairs of inner product Sobolev spaces and Hörmander spaces that form a two-sided refined Sobolev scale. For the latter spaces, the regularity indices are an arbitrary real number and a positive function that varies slowly at infinity in the sense of Karamata. We prove theorems on a priori estimates of generalized solutions to the problem and their regularity.

Keywordsa priori estimate for solutions, elliptic boundary-value problem, Hörmander space, Noetherian operator, regularity of solutions, slowly varying function
References: 
  1. Lawruk B. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 1963, 11, No 5: 257–267 (in Russian).
  2. Lawruk B. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 1963, 11, No 5: 269–278 (in Russian).
  3. Ciarlet P.G. Plates and junctions in elastic multi-structures. An asymptotic analysis, Paris: Masson, 1990.
  4. Nazarov S., Pileckas K. J. Reine Angew. Math., 1993, 438: 103–141.
  5. Kozlov V.A., Maz'ya V.G., Rossmann J. Elliptic boundary value problems in domains with point singularities, Providence: Amer. Math. Soc., 1997.
  6. Roitberg Ya.A. Elliptic boundary value problems in the spaces of distributions, Dordrecht: Kluwer, 1999. https://doi.org/10.1007/978-94-015-9275-8
  7. Roitberg Ya.A. Dokl. Math., 1964, 5: 1034–1038.
  8. Hörmander L. Linear differential operators with partial derivatives, Moscow: Mir, 1965 (in Russian).
  9. Karamata J. Mathematica (Cluj), 1930, 3: 33–48.
  10. Mikhailets V.A., Murach A.A. Ukr. Math. J., 2006, 58, No 3: 398–417. https://doi.org/10.1007/s11253-006-0074-9
  11. Mikhailets V.A., Murach A.A. Banach J. Math. Anal., 2012, 6, No 2: 211–281. https://doi.org/10.15352/bjma/1342210171
  12. Mikhailets V.A., Murach A.A. Hörmander spaces, interpolation, and elliptic problems, Berlin, Boston: De Gruyter, 2014. https://doi.org/10.1515/9783110296891
  13. Chepurukhina I. S. Zb. prats' Inst. Mat. NAN Ukraine, 2014, 11, No 2: 284–304 (in Ukrainian).
  14. Seneta E. Regularly varying functions, Berlin: Springer, 1976. https://doi.org/10.1007/BFb0079658
  15. Volevich L.R., Paneah B. P. Uspekhi Mat. Nauk, 1965, 20, No 1: 3–74 (in Russian).