2017
Том 69
№ 7

All Issues

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

Chepurukhina I. S., Murach A. A.


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.

English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 5, pp 764-784.

Citation Example: Chepurukhina I. S., Murach A. A. Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces // Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 672–691.