Parabolic pseudodifferential initial-boundary value problems in complete scale of spaces
by
Alexander Kozhevnikov
Dept. of Mathematics, University of Haifa, Israel

The aim of the talk is to establish a complete scale of isomorphisms for parabolic pseudodifferential initial-boundary value problems. The pseudodifferentiability of the problem means that it is generated by operators of the Boutet de Monvel calculus.

Parabolic pseudodifferential initial-boundary value problems have been studied by V. Purmonen (1989, !990), G. Grubb and V. Solonnikov (1990) and G. Grubb (1995) as an extension of the results by M. Agranovich and M. Vishik (1963) and J. Lions and E. Magenes (1972) concerning the differential case.

In these papers the parabolic problems have been consideredin the anisotropic Sobolev spaces of functions with s spatial and s/w time derivatives, where the even positive number w is the parabolic weight. More precisely, it has been proved that the operator generated by a parabolic initial-boundary value problem is an isomorphism between the anisotropic Sobolev space of functions "with s spatial and s/w time derivatives" and the space "with s-d spatial and (s-d)/w time derivatives", where d=mw is the order of the differential operator and s >= d.

S. Eidelman and N. Zhitarashu (1998) extended the theorem on isomorphism for parabolic differential problems to a complete scale of spaces, i.e. for any real number s. This theorem is a further elaboration of the results by Ya. Roitberg (1996) conserning the complete scale of isomorphisms for elliptic differential boundary-value problems. The first 5 chapters (170 pages) of the monograph by Eidelman and Zhitarashu (19980 are devoted to a rather complicated and sometimes tedious proof of the theorem on the complete scale of isomorphisms.

The aim of this paper is to establish theorems on the complete scale of isomorphisms for parabolic pseudodifferential initial-boundary value problems.

Due to the fact, that for any parabolic pseudodifferential initial-boundary value problem there exists an inverse operator bbelonging to the Boutet de Monvel calculus, the obtained proof is essentially shorter than in the monograf by Eidelman and Zhitarashu.

Date received: May 31, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-58.