Lipschitz Domains,Domains with Corners,and the Hodge Laplacian

ABSTRACT

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to domains with corners.     

Asymptotics of solutions to the Laplace–Beltrami equation on a rotation surface with a cusp

In this work we study an asymptotic behaviour of solutions to the Laplace–Beltrami operator on a rotation surface near a cuspidal point. To this end we use the WKB-approximation. This approach describes the asymptotic behaviour of the solution more explicitly than abstract theory for operators with operator-valued coefficients.     

The Chern-Connes character for the Dirac operator on manifolds with boundary

A cyclic cocycle is constructed for the Dirac operator on a compact spin manifold with boundary with the -invariant cochain introduced as the boundary correction term. This cocycle is seen to satisfy certain growth condition weaker than being entire and its pairing with the Chern characters of idempotents as well as the relevant index formulae are studied. The -cochain is a generalization of the Atiyah-Patodi-Singer -invariant and it carries information on the APS -invariants for Dirac operators twisted by bundles. It is also shown that one obtains the transgressed Chern character, defined by Connes and Moscovici, by applying the boundary operatorB in the cyclic bicomplex to the higher components of the -cochain.