Green’s Formula and Theorems on Isomorphisms for General Elliptic Problems for Douglis-Nirenberg Elliptic Systems

Abstract

Green’s formula is obtained for general elliptic boundary value problem for elliptic systems of Douglis-Nirenberg structure. The formally adjoint problem with respect to Green’s formula is studied. The same results are obtained for parameter elliptic and parabolic problems for general systems of equations. The Green’s formula permits us to prove various theorems on complete collections of isomorphisms for problems under consideration. Relations between these theorems are studied. Various applications of the isomorphisms theorems are considered. In particular, the question on smoothness of weak solutions of such problems in the whole domain is studied. These investigations were stimulated by works of M.S. Agranovich and A.N. Kozhevnikov in the spectral theory of such problems.1991 AMS Classification35J4035J55