Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case

Abstract

In Hilbert Hörmander spaces, we study the initial-boundary-value problems for arbitrary parabolic differential equations of the second order with Dirichlet boundary conditions or general boundary conditions of the first order in the case where the solutions of these problems belong to the space $H^{2,1,\varphi}$. It is shown that the operators corresponding to these problems are isomorphisms between suitable Hörmander spaces. The regularity of the functions that form these spaces is characterized by a couple of numerical parameters and a functional parameter $\varphi$ slowly varying at infinity in Karamata’s sense. Due to the presence of the parameter $\varphi$, the Hörmander spaces describe the regularity of the functions more precisely than the anisotropic Sobolev spaces.