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

Abstract

In this work, it is considered an elliptic operator L of mth order with non-smooth coefficients in a non-standard grand Sobolev space W-q)(m) (Omega) on a bounded domain Omega subset of R-n generated by the norm of the grand Lebesgue space L-q) (Omega). Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in M-q)(m)(Omega) 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 Omega.