Probability Seminar: Phanuel Mariano

Geometric properties of optimizers for the maximum gradient of the torsion function. The torsion function for a domain is a classical object in the theory of elasticity that measures the resistance to twisting for a rod with cross-sections given by such domain. Probabilistically, it is given by the mean exit time of Brownian motion from a domain as a function of its starting point. Motivated by this, we consider an extremal problem for the maximum gradient of the torsion function, ranging over all bounded convex planar domains. We show that this extremal problem admits a maximizer, and that any maximizer has a C^1 boundary containing a line segment on which its gradient attains its maximum. The proofs are probabilistic in nature and use novel coupling techniques. This talk is based on joint work with Krzysztof Burdzy and Ilias Ftouhi.