Ivan G. Avramidi (Department of Mathematics, The University of Iowa)

ABSTRACT:

We study a Laplace type partial differential operator acting on sections of a vector bundle over a compact Riemannian manifold with smooth boundary with the Gilkey-Smith boundary conditions, which involve both the normal and tangential derivatives on the boundary. This boundary value problem becomes strongly elliptic only under a certain condition on the boundary operator. We find a simple criterion of strong ellipticity and compute the first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel. We also obtain the local leading asymptotics of the heat kernel diagonal and show that in the non-elliptic case the heat-kernel diagonal is non-integrable near the boundary, which reflects the fact that the heat kernel is not of trace class.
 

Slides of the lecture: PostScript

Preprint: math-ph/9812010