Isoperimetric Inequalities for the Cauchy-Dirichlet Heat Operator

Abstract

In this paper we prove that the firsts-number of the Cauchy-Dirichlet heat operator is minimized in a circular cylinder among all Euclidean cylindric domains of a given measure. It is an analogue of the Rayleigh-Faber-Krahn inequality for the heat operator. We also prove a Hong-Krahn-Szego and a Payne- ¨ Polya-Weinberger type inequalities for the Cauchy-Dirichlet heat operator.