About the maximization of the gradient of the torsion function

In this talk, we address the problem of maximizing the supremum norm of the gradient of the torsion function over planar convex domains with fixed measure (or perimeter). We prove that an optimal shape exists, its boundary is C^1 regular, and it features a straight segment on which the gradient is maximal. The proofs combine probabilistic and analytic methods. The talk is based on a joint work with Krzysztof Burdzy (University of Washington) and Phanuel Mariano (Union College).