General elliptic boundary-value problems in Hörmander—Roitberg spaces

TitleGeneral elliptic boundary-value problems in Hörmander—Roitberg spaces
Publication TypeJournal Article
Year of Publication2018
AuthorsKasirenko, TM
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2018.02.003
Issue2
SectionMathematics
Pagination3-11
Date Published2/2018
LanguageUkrainian
Abstract

We prove theorems on the character of solvability and regularity of solutions of general elliptic boundary-value problems in Hilbert Hörmander spaces modified by Roitberg. An arbitrary real number and a sufficiently general weight function of frequency variables serve as the indices of regularity for these spaces.

Keywordsa priori estimate, elliptic problem, Fredholm operator, Hörmander space, regularity of a solution, RO-varying function
References: 
  1. Lions, J.-L. & Magenes, E. (1972). Non-homogeneous boundary-value problems and applications. Vol. 1. New York, Heidelberg: Springer.
  2. Roitberg, Ya. A. (1996). Elliptic boundary value problems in the spaces of distributions. Dordrecht: Kluwer Acad. Publishers. doi: https://doi.org/10.1007/978-94-011-5410-9
  3. Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer. doi: https://doi.org/10.1007/978-3-642-46175-0
  4. Mikhailets, V. A. & Murach, A. A. (2014). Hörmander spaces, interpolation, and elliptic problems. Berlin, Boston: De Gruyter. doi: https://doi.org/10.1515/9783110296891
  5. Agranovich, M. S. (1997). Elliptic boundary problems. Encycl. Math. Sci. Vol. 79. Partial differential equations, IX. Berlin: Springer. doi: https://doi.org/10.1007/978-3-662-06721-5_1
  6. Kozlov, V. A., Maz'ya, V. G. & Rossmann, J. (1997). Elliptic boundary value problems in domains with point singularities. Providence: Amer. Math. Soc.
  7. Seneta, E. (1976). Regularly varying functions. Berlin: Springer. doi: https://doi.org/10.1007/BFb0079658
  8. Volevich, L. R. & Paneah, B. P. (1965). Certain spaces of generalized functions and embedding theorems. Russ. Math. Surveys, 20, No. 1, pp. 1-73. doi: https://doi.org/10.1070/RM1965v020n01ABEH004139
  9. Mikhailets, V. A. & Murach, A. A. (2006). Refined scales of spaces and elliptic boundary-value problems. II. Ukr. Math. J., 58, No. 3, pp. 398-417. doi: https://doi.org/10.1007/s11253-006-0074-9
  10. Mikhailets, V. A. & Murach, A. A. (2013). Extended Sobolev scale and elliptic operators. Ukr. Math. J., 65, No. 3, pp. 435-447. doi: https://doi.org/10.1007/s11253-013-0787-5
  11. Roitberg, Ja. A. (1964). Elliptic problems with non-homogeneous boundary conditions and local increase of smoothness of generalized solutions up to the boundary. Soviet. Math. Dokl., 5, pp. 1034-1038.
  12. 1Mikhailets, V. A. & Murach, A. A. (2008). An elliptic boundary-value problem in a two-sided refined scale of spaces. Ukr. Math. J., 60, No. 4, pp. 574-597. doi: https://doi.org/10.1007/s11253-008-0074-z
  13. Roitberg, Ja. A. (1970). Homeomorphism theorems and Green's formula for general elliptic boundary value problems with boundary conditions that are not normal. Sb. Math., 12, No. 2, pp. 177-212. doi: https://doi.org/10.1007/BF01085380
  14. Kostarchuk, Ju. V. & Roitberg, Ja. A. (1973). Isomorphism theorems for elliptic boundary value problems with boundary conditions that are not normal. Ukr. Math. J., 25, No. 2, 222-226. doi: https://doi.org/10.1007/BF01096983
  15. Kostarchuk, Ju. V. (1973). Local increase of the smoothness of generalized solutions to elliptic boundary value problems with boundary conditions that are not normal. Ukr. Mat. Zh., 25, No. 4, 536–540 (in Russian). doi: https://doi.org/10.1007/BF01096983