Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol

Zeta functions of elliptic cone operators

This paper is an overview of aspects of the singularities of the zeta function, equivalently, of the small time asymptotics of the trace of the heat semigroup, of elliptic cone operators. It begins with a brief description of classical results for regular differential operators on smooth manifolds, and includes a concise introduction to the theory of cone differential operators. The later secti...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators

Heat Kernel Asymptotics for Laplace Type Operators and Matrix Kdv Hierarchy

We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results obtained earlier for the scalar Laplacian and present closed formulas for all heat invariants associated with these operators. As another application, we present ...

Asymptotics of the heat equation with ‘exotic’ boundary conditions or with time dependent coefficients

The heat trace asymptotics are discussed for operators of Laplace type with Dirichlet, Robin, spectral, D/N, and transmittal boundary conditions. The heat content asymptotics are discussed for operators with time dependent coefficients and Dirichlet or Robin boundary conditions.

Leading Terms in the Heat Invariants 3

Let D be a second-order diierential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on D are given by homogeneous, invariant, local formulas. Within the set of allowable expressions of a given homogeneity there is a ltration by degree, in which elements of the smallest class have the highest degree. Modulo quadra...