Elliptic problems with boundary conditions of high orders in H\"ormander spaces

TL;DR

This paper studies elliptic boundary value problems within H"ormander spaces with high-order boundary conditions, establishing boundedness, Fredholm properties, and regularity results for solutions in these generalized function spaces.

Contribution

It introduces a framework for elliptic problems with high-order boundary conditions in H"ormander spaces, proving boundedness, Fredholmness, and regularity theorems.

Findings

Operator is bounded and Fredholm on H"ormander spaces.

Existence of isomorphism for the elliptic problem operator.

Local regularity and continuity conditions for solutions.

Abstract

In a class of inner product H\"ormander spaces, we investigate a general elliptic problem for which the maximum of orders of boundary conditions is grater than or equal to the order of elliptic equation. The order of regularity for these spaces is an arbitrary radial positive function RO-varying at infinity in the sense of Avakumovi\'c. We prove that the operator of the problem under investigation is bounded and Fredholm on appropriate pairs of H\"ormander spaces indicated. A theorem on isomorphism generated by this operator is proved. For generalized solutions to this problem, we establish a local a priory estimate and prove a theorem about their local regularity in H\"ormander spaces. As application, we obtain new sufficient conditions under which given derivatives of the solutions are continuous.