On solvability in the small and Schauder-type estimates for higher order elliptic equations in grand Sobolev spaces (nonseparable case)


Bilalov B. T. , Zeren Y. , Sadigova S. R. , Cetin S.

Applicable Analysis, 2022 (Journal Indexed in SCI Expanded) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2022
  • Doi Number: 10.1080/00036811.2022.2052859
  • Journal Name: Applicable Analysis
  • Journal Indexes: Science Citation Index Expanded, Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: Elliptic operator, Schauder estimates, grand Sobolev and grand Lebesgue spaces, PIECEWISE-LINEAR PHASE, MORREY SPACES, INTEGRAL-OPERATORS, SYSTEM, REGULARITY, EXPONENTS, BOUNDARY, THEOREMS, BASICITY, LEBESGUE

Abstract

© 2022 Informa UK Limited, trading as Taylor & Francis Group.In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space (Formula presented.) on a bounded domain (Formula presented.) generated by the norm of the grand Lebesgue space (Formula presented.). Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in (Formula presented.) and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω.