Leander Geisinger, Timo Weidl
Universal Bounds for Traces of the Dirichlet Laplace Operator
Preprint series: ESI preprints

MSC:

35P15 Estimation of eigenvalues, upper and lower bounds

35Q80 Applications of PDE in areas other than physics

Abstract: We derive upper bounds for the trace of the heat kernel
$Z(t)$ of the Dirichlet Laplace operator in an open set
$\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using
the same methods we give bounds on $Z(t)$ in domains of infinite volume.

For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains
the sharp first term and a correction term reflecting
the properties of the short time asymptotics of $Z(t)$.
To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.