Elliptic problems with boundary operators of higher orders in H\"ormander-Roitberg spaces

TL;DR

This paper studies elliptic boundary-value problems with high-order boundary operators in specialized H"ormander-Roitberg spaces, proving boundedness, Fredholm properties, and regularity of solutions within these advanced functional frameworks.

Contribution

It introduces a new analysis of elliptic problems with boundary operators of higher order using H"ormander-Roitberg spaces, establishing boundedness, Fredholmness, and regularity results.

Findings

Operator is bounded and Fredholm between specific Hilbert spaces.

Established local a priori estimates for solutions.

Derived conditions for solutions to have classical derivatives.

Abstract

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic H\"ormander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the boundary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order.