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 $$W_{q)}^{2} \left( \varOmega \right) $$ on a bounded n-dimensional domain $$\varOmega \subset R^{n} $$ with a sufficiently smooth boundary $$\partial \varOmega $$ , generated by the norm of the grand-Lebesgue space $$L_{q)}\left( \varOmega \right) $$ . These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace $$N_{q)}^{2} \left( \varOmega \right) $$ is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion $$W_{q}^{2} \left( \varOmega \right) \subset N_{q)}^{2} \left( \varOmega \right) $$ holds, where $$W_{q}^{2} \left( \varOmega \right) $$ is the classical Sobolev space. This raises specific questions dictated by the theory of spaces $$W_{q}^{2} \left( \varOmega \right) $$ , for example, the characterization of the space of traces of functions from $$N_{q)}^{1} \left( \varOmega \right) $$ cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from $$N_{q)}^{k} \left( \varOmega \right) $$ 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 $$N_{q)}^{2} \left( \varOmega \right) $$ 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 .