On a Complete Scale of Isomorphisms for Elliptic and Parabolic Pseudodifferential Boundary-Value Problems

Alexander Kozhevnikov Dept. of Math., Univ. of Haifa, Israel

In monographs by J.-L.Lions and E.Magenes (1968) and by Ya.Roitberg (1996) theorems on complete scales of isomorphisms have been established which, roughly speaking, mean that operators generated by elliptic differential boundary-value problems are isomorghisms (Fredholm operators) between Sobolev-type spaces of functions with s and s-d derivatives, where d is the order of the the elliptic operator. The completeness of the scale means that s can be an arbitrary real number.

In a monogragh by S.Eidelman and N.Zhitarashu (1990) a theorem on a complete scale of isomorphisms has been obtained for parabolic differential initial boundaryvalue problems.

Due to the fact, that for elliptic and parabolic pseudodifferential initial boundaryvalue problems there exist parametrices belonging to the Boutet de Monvel algebra, a much shorter proof has been found as well as some applications of the results to the spectral theory.