Analysis on Manifolds with Geometric Singularities
by
Bert-Wolfgang Schulze
University of Potsdam, Germany

The properties of solutions to partial differential equations on such a configuration are very much affected by the nature of singularities.

In the simplest case of conical singularities ellipticity is determined by the bijectivity of two principal symbolic components, namely the interior (scalar) symbol of Fuchs type (in stretched coordinates) and the (operator-valued) conormal symbol which is a meromorphic operator family acting between Sobolev spaces on the base of the cone. Solutions in weighted Sobolev spaces have asymptotics, contributed by the poles of the inverse conormal symbol (known by Kondratyev's work). Another example are manifolds with edges, locally modelled by wedges. Ellipticity consists of the bijectivity of the principal interior (scalar) symbol (which is edge-degenerate in stretched coordinates) and the (operator-valued) principal edge symbol. The latter one is a family of operators on the (infinite) model cone of local wedges. It determines elliptic edge conditions, analogously to boundary conditions in boundary value problems, here (in simplest cases) with an analogue of the Shapiro-Lopatinskij condition. Manifolds with boundary belong to the category of manifolds with edges, where the edge is the boundary and the inner normal the model cone.

The analysis of elliptic (and also parabolic) operators on manifolds with higher singularities is governed by hierarchies of principal symbols and additional edge conditions on the lower-dimensional strata of the configuration. The program of the calculus consists of the construction of parametrices, the characterisation of weighted Sobolev spaces (in general, with multiple weights, and anisotropic in parabolic cases) and subspaces with (in general, iterated) asymptotics to describe the regularity of solutions.

The lecture presents a new approach of iteratively applying cone and wedge constructions on the level of pseudo-differential algebras with corresponding symbolic hierarchies.

The ideas are also motivated by models of applied sciences with singular geometries, e.g., of crack theory or mixed problems, where the boundary conditions have discontinuities. Moreover, there are new relations to the index theory with interesting challenges and open problems in connection with ``higher'' versions of the Atiyah-Singer index theorem.

References:

[1 ] D. Kapanadze and B.-W. Schulze, Crack theory and edge singularities. Kluwer Academic Publ., Dordrecht, 2003, (to appear).
[2 ] T. Krainer and B.-W. Schulze, On the inverse of parabolic systems of partial differential equations of general form in an infinite space-time cylinder. Advances in Partial Differential Equations (Parabolicity, Volterra Calculus, and Conical Singularities) (S. Albeverio, M. Demuth, E. Schrohe, and B.-W. Schulze, eds.), Oper. Theory Adv. Appl., vol. 138, Birkhäuser Verlag, Basel, 2002, pp. 93-278.
[3 ] B.V. Fedosov, B.-W. Schulze and N.N. Tarkhanov, Analytic index formulas for elliptic corner operators. Ann. Inst. Fourier 52, 3 (2002), 899-982.
[4 ] V. Nazaikinskij, A. Savin, B.-W. Schulze and B. Ju. Sternin, Elliptic theory on manifolds with nonisolated singularities: II. Products in elliptic theory on manifolds with edges. Preprint 2002/15, Institut für Mathematik, Potsdam, 2002.
[5 ] B.-W. Schulze, Boundary value problems and singular pseudo-differential operators. J. Wiley, Chichester, 1998.
[6 ] B.-W. Schulze, Operators with symbol hierarchies and iterated asymptotics. Publications of RIMS, Kyoto University, 38, 4 (2002), 735-802.

Date received: May 5, 2003