Faber-Krahn type inequalities and uniqueness of positive solutions on metric measure spaces

Abstract

We consider a general class of metric measure spaces equipped with a strongly local regular Dirichlet form and provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of the classical Lieb's inequality, and (ii) uniqueness of nonnegative super-solutions to semi-linear elliptic equations on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [28] to local and non-local Dirichlet spaces.