Interior Schauder-Type Estimates for Higher-Order Elliptic Operators in Grand-Sobolev Spaces

Abstract

In this paper  an elliptic operator of the $m$-th order  $L$ with continuous coefficients in the $n$-dimensional domain $Omega subset R^{n} $ in the non-standard Grand-Sobolev space $W_{q)}^{m} left(Omega right), $ generated by the norm $left| , cdot , right| _{q)} $ of the Grand-Lebesgue space $L_{q)} left(Omega right), $ is considered.  Interior  Schauder-type estimates  play a very important role in solving the Dirichlet problem for the equation $Lu=f$. The considered non-standard spaces are not separable, and therefore, to use classical methods for treating solvability problems in these spaces, one needs to modify these methods. To this aim, based on the shift operator, separable subspaces of these spaces are determined, in which finite infinitely differentiable functions are dense.  Interior  Schauder-type estimates  are established with respect to these subspaces. It should be noted that Lebesgue spaces $L_{q} left(Gright), $ are strict   parts of these subspaces. This work is a continuation of the authors  of the work cite{28}, which established the solvability in the small of higher order elliptic equations in grand-Sobolev spaces.