On properties of the asymptotic expansion of the heat trace for the N/D problem

TL;DR

This paper investigates the heat trace asymptotics for a spectral problem with mixed boundary conditions, showing that a classical short-time asymptotic expansion with fractional powers does not exist.

Contribution

It demonstrates the non-existence of a classical asymptotic expansion with fractional powers for the heat trace in the mixed Dirichlet-Neumann boundary problem.

Findings

No classical asymptotic expansion with fractional powers exists

The behavior of the heat trace differs from standard cases

Implications for spectral geometry and boundary value problems

Abstract

The spectral problem where the field satisfies Dirichlet conditions on one part of the boundary of the relevant domain and Neumann on the remainder is discussed. It is shown that there does not exist a classical asymptotic expansion for short time in terms of fractional powers of $t$ with locally computable coefficients.