Interior Schauder-Type Estimates for Higher-Order Elliptic Operators in Grand-Sobolev Spaces (original) (raw)

In this paper an elliptic operator of the mmm-th order LLL with continuous coefficients in the nnn-dimensional domain OmegasubsetRnOmega subset R^{n} OmegasubsetRn in the non-standard Grand-Sobolev space Wq)mleft(Omegaright),W_{q)}^{m} left(Omega right), Wq)mleft(Omegaright), generated by the norm left∣,cdot,right∣q)left| , cdot , right| _{q)} left∣,cdot,right∣q) of the Grand-Lebesgue space Lq)left(Omegaright),L_{q)} left(Omega right), Lq)left(Omegaright), is considered. Interior Schauder-type estimates play a very important role in solving the Dirichlet problem for the equation Lu=fLu=fLu=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 Lqleft(Gright),L_{q} left(Gright), Lqleft(Gright), are strict parts of these subspaces. This work is...