Fredholmness of the Dirichlet Problem for 2mth-Order Elliptic Equations in Grand Sobolev Spaces

Bilalov B. T., Sadigova S. R., Nasibova N. P.

Mathematical Methods in the Applied Sciences, 2025 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2025
  • Doi Number: 10.1002/mma.11098
  • Journal Name: Mathematical Methods in the Applied Sciences
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Applied Science & Technology Source, Communication Abstracts, Compendex, INSPEC, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Keywords: 2mth-order elliptic equation, Fredholmness, grand Sobolev spaces, strong solution
  • Yıldız Technical University Affiliated: No

Abstract

In this paper, on a bounded domain (Formula presented.) with a sufficiently smooth boundary (Formula presented.), it is considered a uniformly elliptic equation of (Formula presented.) th order, the coefficients of which are continuous in the principal part. Grand Lebesgue space (Formula presented.), is studied. This space is nonseparable, and it is defined a separable subspace (Formula presented.) of (Formula presented.), in which infinitely differentiable functions are dense. Furthermore, the grand Sobolev space (Formula presented.) of (Formula presented.) th-order differentiable in the Sobolev sense functions is introduced. This space is generated by the subspace (Formula presented.). For the given equation, a Schauder-type estimate up to the boundary is proved. Using this estimate, we establish an a priori estimate and then the Fredholmness of the (Formula presented.) th-order elliptic equation under consideration in (Formula presented.). By a solution, we mean a strong solution.