In this talk, we consider the functional\[J(\Omega) = \frac{\|\nabla u_\Omega\|_\infty}{\sqrt{|\Omega|}},\]where $\Omega$ is an open subset of the plane, $u_\Omega$ is its torsion function, and $|\Omega|$ denotes its area.
We prove that the functional $J$ admits a maximizer among convex planar domains. We then show that any optimal domain is regular (of class $C^1$) and that its boundary contains a segment on which the function $|\nabla u_\Omega|$ attains its maximum value. The proofs rely on probabilistic approaches, whose main intuitions and usefulness will be highlighted. Finally, we present numerical simulations and state some conjectures.
This talk is based on works in collaboration with Krzysztof Burdzy (University of Washington, USA), Chiu-Yen Kao (Claremont McKenna College, USA), Xuefeng Liu (Tokyo Woman's Christian University, Japan), and Phanuel Mariano (Union College, USA).
