Fredholmness of the Dirichlet Problem for 2mth-Order Elliptic Equations in Grand Sobolev Spaces

Abstract

In this paper, on a bounded domain Omega subset of R-n with a sufficiently smooth boundary partial derivative Omega, it is considered a uniformly elliptic equation of 2mth order, the coefficients of which are continuous in the principal part. Grand Lebesgue space L-p)(Omega), 1 < +infinity, is stud-ied. This space is nonseparable, and it is defined a separable subspace Np)(Omega) of Lp)(Omega), in which infinitely differentiable functions are dense. Furthermore, the grand Sobolev space N-p)(2m)(Omega)of 2mth-order differentiable in the Sobolev sense functions is introduced. This space is generated by the subspace N-q)(Omega). For the given equation, a Schauder-type estimate up to the boundary isproved. Using this estimate, we establish an a priori estimate and then the Fredholmness of the 2mth-order elliptic equation under consideration in N-p)(2m)(Omega). By a solution, we mean a strong solution.