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

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(G\right)\, are strict   parts of these subspaces. This work is a continuation of the authors  of the work \cite{۲۸}, which established the solvability in the small of higher order elliptic equations in grand-Sobolev spaces.

Keywords:

Elliptic operator , Higher-order , Interior Schauder-type Estimates , Grand-Sobolev space