Coercivity condition for overdetermined boundary values (Q579526)

The purpose of this paper is to develop the coercivity condition of Lopatinskij's type given in the previous papers of the first author [Sov. Math., Dokl. 25, 139-144 (1982); translation from Dokl. Akad. Nauk SSSR 262, 810-814 (1982; Zbl 0537.35059); and Ukr. Math. J. 36, No.3, 305-310 (1984); translation from Ukr. Mat. Zh. 36, No.3, 340-346 (1984; Zbl 0595.35081)]. In fact, if M is a smooth manifold with a smooth boundary \(\Gamma\) one considers pairs (A,B) of differential operator A and local linear boundary operator B acting on the sections of vector bundles over M and respectively over \(\Gamma\). The mentioned condition gives the exactness of the complex of finite-dimensional bundles over the boundary \(\Gamma\), but in the previous papers his formulation was in some sense very complicated. In the present paper this condition is simplified and in some particular cases (including large classes of ''non-degenerate'' operators) it is formulated only in terms of the pair (A,B). One can use the obtained theorems to introduce an Euler characteristic for the boundary problem of type (A,B), which, following the authors' opinion, will play for the overdetermined problems the same role as the notion of index for the boundary problems without overdeterminicity.