Let Ω ⊂ R n be a convex domain and let f : Ω → R be a positive, subharmonic function (i.e. ∆f ≥ 0). Thenwhere cn ≤ 2n 3/2 . This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies cn ≥ n − 1. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other Ω 2 ⊂ Ω 1 ⊂ R n :

We study the relationship between the geometry of smoothly bounded domains in complete Riemannian manifolds and the associated sequence of L 1 -norms of exit time moments for Brownian motion. We establish bounds for Dirichlet eigenvalues and, for closed manifolds, we establish a comparison result for elements of the moment sequence using lower bounds on Ricci curvature.

We give upper bounds on the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold; the bounds involve L1‐norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Pólya. We also prove lower bounds for Dirichlet eigenvalues using invariants that arise during the examination of the relationship between the heat content and exit time moments.

Let γ be a smooth, non-closed, simple curve whose image is symmetric with respect to the y-axis, and let D be a planar domain consisting of the points on one side of γ, within a suitable distance δ of γ. Denote by µ odd 1 (D) the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the y-axis. If γ satisfies some simple geometric conditions, then µ odd 1 (D) can be sharply estimated from below in terms of the length of γ, its curvature, and δ. Moreover, we give explicit conditions on δ that ensure µ odd 1 (D) = µ 1 (D). Finally, we can extend our bound on µ odd 1 (D) to a certain class of three-dimensional domains. In both the two-and three-dimensional settings, our domains are generically non-convex.2010 Mathematics Subject Classification. 35J25,35P15.

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