Rayleigh–Faber–Krahn and Hong–Krahn–Szegö-type inequalities for the absolute value of the heat operator

Abstract

The main aim of this paper is to show that the first singular number of the generalized Cauchy–Dirichlet heat operator is minimized by a circular cylinder among all domains of the same measures with the circular cylinder in Euclidean space. This result gives us an analogue of the celebrated Rayleigh–Faber–Krahn inequality for the absolute value of the heat operator. Further, we obtain an analogue of the Hong–Krahn–Szegö inequality for the same operator.