Universal bounds for traces of the Dirichlet Laplace operator

Geisinger, L.; Weidl, T.
October 2010
Journal of the London Mathematical Society;Oct2010, Vol. 82 Issue 2, p395
Academic Journal
We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set Ω ⊂ ℝd, with d ≥ 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.


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