Advanced Course on Pseudodifferential Operators and

Their Applications 2

Date: 16-25 September, 2002

Location: TICMI (Tbilisi)

B.-Wolfgang Schulze (University of Potsdam, Germany)

THE PSEUDODIFFERENTIAL CALCULUS FOR SINGULAR AND DEGENERATE OPERATORS

Summary: The pseudo-differential calculus of boundary value problems under the aspect of ellipticity and parametrix constructions is characterized by a principal symbolic hierarchy, consisting in this case by the pair of principal interior and boundary symbols. In particular, the boundary symbolic component is a family of operators, acting in Sobolev spaces on the half-axis in normal direction to the boundary.

According the specific interior operators that may be with or without the transmission property, the boundary symbols take values in various subalgebras of the cone algebra on the half-axis, where the half-axis is regarded as the simplest example of a space with conical singularities. The discussion so far concerns domains or manifolds with smooth boundary. If the boundary has edges or corners, the structure of associated pseudo-differential operators is more rich. The principal symbolic hierarchy consists of a larger tuple of components, most of them operator-valued, operating on spaces with lower order of singularity (e.g., cones). Those symbols have again a subordinate symbolic structure.

Configurations with edges and corners are of interest in applications to models in mechanics, in particular, connected with crack problems or mixed and transmission problems. A challenge for the analysis is to establish adequate operator algebras with symbolic structures that admit the construction of parametrices and to characterize asymptotics of solutions. The series of lectures gives an introduction and an overview of this fascinating field that has a long tradition in the analysis of singular integral operators and that has seen an enormous development in the past few years, with many relations also to geometry and topology.

Coordinator: George Jaiani

This course is suitable for advanced graduate students or recent Ph.D.'s. The participants will also have an opportunity to give 20-minute talks on their own work at a mini-symposium which will take place during the Advanced Course. Lectures and abstracts of the talks will be published and distributed among the lecturers and participants after Advanced Course. The registration fee for participants is 400 USD which includes all local expenses during the Advanced Course. A restricted number of participants will be awarded grants.

Further information: TICMI, I.Vekua Institute of Applied Mathematics of Tbilisi State University, University Str. 2, Tbilisi 380043, Georgia

e.mail: jaiani@viam.hepi.edu.ge

Tel.:+995 32 305995