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

2018/10/26 by Anup Biswas, Biswas, Anup, Janna Lierl +1
Mathematics · #Functional Equations Stability Results #Geometric Analysis and Curvature Flows #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1810.11577

Abstract

We consider a general class of metric measure spaces equipped with a regular\nDirichlet form and then provide a lower bound on the hitting time probabilities\nof the associated Hunt process. Using these estimates we establish (i) a\ngeneralization of the classical Lieb's inequality on metric measure spaces and\n(ii) uniqueness of nonnegative super-solutions on metric measure spaces.\nFinally, using heat-kernel estimates we generalize the local Faber-Krahn\ninequality recently obtained in [LS18].\n

Related

• A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces
• On notions of harmonicity
• Symmetrization and sharp Sobolev inequalities in metric spaces
• Sharp uncertainty principles on metric measure spaces
• Stability of heat kernel estimates for symmetric non-local Dirichlet forms