Speaker: Phanuel Mariano (Union College)
Title: Sharp bounds for the torsional rigidity and the first Dirichlet eigenvalue of triangles and rectangles
Abstract: We consider a functional involving the product of the torsional rigidity and the first Dirichlet eigenvalue of a planar domain normalized by its area. This functional was originally considered by Pólya who first showed that this quantity is bounded by 1. It has been conjectured that this functional is bounded above and below by π^2/12 and π^2/24 over the class of bounded planar convex domains. We prove this is true for the class of triangles and rectangles. In particular, we prove precise estimates for triangles and a monotonicity property for rectangles. This is joint work with Rodrigo Bañuelos.
