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

TL;DR

This paper extends classical inequalities and uniqueness results for positive solutions to metric measure spaces with Dirichlet forms, using probabilistic and heat kernel techniques to generalize known inequalities.

Contribution

It introduces generalized Faber-Krahn inequalities and uniqueness results for super-solutions on metric measure spaces, broadening classical Euclidean results.

Findings

Lower bounds on hitting time probabilities for Hunt processes

Generalization of Lieb's inequality to metric measure spaces

Uniqueness of nonnegative super-solutions in this setting

Abstract

We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then 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 on metric measure spaces and (ii) uniqueness of nonnegative super-solutions on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [LS18].