Elliptic problems with boundary conditions of higher orders in Hörmander spaces

Abstract

In a class of inner product H¨ormander spaces, we study a general elliptic problem for which the maximum order of the boundary conditions is not smaller than the order of the elliptic equation. The role of the order of regularity of these spaces is played by an arbitrary radial positive function $R_O$-varying at infinity in the sense of Avakumovi´c. We prove that the operator of the problem under investigation is bounded and Fredholm on the appropriate pairs of the indicated H¨ormander spaces. A theorem on isomorphism generated by this operator is proved. For the generalized solutions of this problem, we establish a local a priori estimate and prove the theorem on the local regularity of these solutions in H¨ormander spaces. As an application, we establish new sufficient conditions of continuity for the given generalized derivatives of the solutions.