Symbolic calculus for boundary value problems on manifolds with edges (Q1865897)

The authors consider boundary value problems for differential operators on a manifold with edges, the aim being to construct an algebra of pseudo-differential operators containing parametrices for the elliptic equations. The case without boundary was studied by \textit{J. B. Gil}, \textit{B. W. Schulze} and \textit{J. Seiler} [Osaka J. Math. 37, No. 1, 221-260 (2000; Zbl 1005.58010)]. In the present paper similar results are obtained for boundary problems. Basic ingredients of the calculus are Mellin pseudo-differential operators with operator-valued amplitude functions in Fréchet spaces. Ellipticity is then expressed as invertibility of Fredholm families, playing the role of conormal symbols for elliptic operators on a cone. As the authors observe, the parameter-dependent operators treated here are useful also for other applications on manifolds with conical singularities, for example computation of heat trace expansion and long-time asymptotics to parabolic boundary value problems.