On the fredholmness of the Dirichlet problem for a second-order elliptic equation in grand-Sobolev spaces

In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes W-q)(2) (Omega) on a bounded n-dimensional domain Omega subset of R-n with a sufficiently smooth boundary partial derivative Omega, generated by the norm of the grand-Lebesgue space L-q) (Omega). These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace N-q)(2) (Omega) is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion W-q(2) (Omega) subset of N-q)(2) (Omega) holds, where W-q(2) (Omega) is the classical Sobolev space. This raises specific questions dictated by the theory of spaces W-q(2) (Omega), for example, the characterization of the space of traces of functions from N-q)(1) (Omega) cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from N-q)(k) (Omega) are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces N-q)(2) (Omega) with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the L-p-theory. This work is a continuation of the research carried out by the authors in articles (Bilalov and Sadigova in Complex Var Elliptic Equ, 2020. https://doi.org/10.1080/17476933.2020.1807965; Bilalov and Sadigova in Sahand Commun Math Anal, 2021. https://doi.org/10.22130/scma.2021.521544.893.