09-155 L. Geisinger, T. Weidl

Universal Bounds for Traces of the Dirichlet Laplace Operator (571K, Postscript) Sep 7, 09

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Abstract. We derive upper bounds for the trace of the heat kernel Z(t) of the Dirichlet Laplace operator in an open set. The result improves an inequality of Kac and is applicable to any open set with finite volume. The bound decays exponentially as t tends to infi nity and it contains the sharp fi rst term and a correction term refl ecting the properties of the short time asymptotics of Z(t). To prove the result we employ refi ned Berezin-Li-Yau inequalities for the Riesz means of the Dirichlet Laplace operator. Using this method we also give bounds on Z(t) in domains of in finite volume.

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