# Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces

### Abstract

We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in appropriate couples of the inner product isotropic Hörmander spaces $H^{s,φ}$, which form the refined Sobolev scale. The order of differentiation for these spaces is given by a real number $s$ and a positive function $φ$ slowly varying at infinity in Karamata’s sense. We consider this problem for an arbitrary elliptic equation $Au = f$ in a bounded Euclidean domain $Ω$ under the condition that $u ϵ H^{s,φ} (Ω),\; s < \text{ord} A$, and $f ϵ L_2 (Ω)$. We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.
Published

25.05.2015

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 67, no. 5, May 2015, pp. 672–691, http://umj.imath.kiev.ua/index.php/umj/article/view/2014.

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Section

Research articles