Elliptic boundary-value problems in the sense of Lawruk on Sobolev and H\"ormander spaces

TL;DR

This paper studies elliptic boundary-value problems with additional boundary unknowns within the framework of H"ormander spaces, proving boundedness, Fredholm property, and regularity of solutions in refined Sobolev scales.

Contribution

It establishes the boundedness and Fredholm property of the operator in H"ormander spaces and analyzes regularity of solutions for Lawruk-type elliptic problems.

Findings

Operator is bounded and Fredholm on H"ormander spaces

Solutions satisfy a priori estimates

Regularity results for generalized solutions

Abstract

We investigate elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on appropriate couples of the inner product isotropic H\"ormander spaces $H^{s,\varphi}$, which form the refined Sobolev scale. The order of differentiation for these spaces is given by the real number $s$ and positive function $\varphi$ that varies slowly at infinity in the sense of Karamata. We consider this problem for an arbitrary elliptic equation $Au=f$ on a bounded Euclidean domain $\Omega$ under the condition that $u\in H^{s,\varphi}(\Omega)$, $s<\mathrm{ord}\,A$, and $f\in L_{2}(\Omega)$. We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.