A. V. Anop

We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at $+\infty$. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.