Christian Rose: Gaussian upper heat kernel bounds and Faber-Krahn inequalities on graphs with unbounded geometry

Abstract:

On Riemannian manifolds it is known that Faber-Krahn inequalities in balls are equivalent to the conjunction of Gaussian upper bounds and volume doubling. Due to the different short-time behaviour of heat kernels on graphs the equivalence is expected to hold on large balls only in this setting. In the case of the normalizing measure, i.e., bounded Laplacian, it turns out that an additional regularity condition on the measure is necessary for the equivalence to hold. If a generalization and unification to arbitrary measure, i.e., possibly unbounded Laplacians, is desired, a new natural local regularity condition enters the equivalence. Moreover, the dimension of the Faber-Krahn inequalities in balls has to vary and relates to the doubling dimension and the growth of vertex degree inside balls.