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 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 [30].for some x ∈ R d where B(x, r) denotes the ball of radius r around x. The above inequality can be seen as a finer version of the classical Faber-Krahn inequality. This inequality has been extended

Liouville type results for systems of equations involving fractional Laplacian in exterior domains

“…(b) Suppose that U (x) |x| m , m > −α, and V (x) |x| n , n > −β. Such potential functions are considered in [4,9,13,25]. It is easy to see that we have Φ U (r) r d+m and Φ V (r) r d+n .…”

“…Such potential functions are considered in [4,9,13,25]. It is easy to see that we have Φ U (r) r d+m and Φ V (r) r d+n .…”

“…In this article we propose a simple and unified probabilistic approach which is capable to deal with such problems in the exterior domains for a large family of f, g. We refer the readers to the discussion at the end of Section 2 to compare the sharpness of our results to the existing literature. Recently, a similar probabilistic approach has been used in [4] for studying Liouville type properties for local Dirichlet forms on metric measure spaces.…”

Liouville theorems for semilinear differential inequalities on sub-Riemannian manifolds

“…in R + × H n , where u 0 , u 1 ∈ L 1 loc (H n ). For more related works on this type of nonexistence theorems on Heisenberg groups, we refer to [2,36,37], for H-type groups [8], for Carnot groups [12,26], and for metric measure spaces [7].…”