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

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.