Geometric properties of optimizers for the maximum gradient of the torsion function

Abstract

Consider $J(\Omega ):=\Vert \nabla u_{\Omega }{\Vert }_{\infty }/\sqrt{|\Omega |}$ and $J_P(\Omega ):=\Vert \nabla u_{\Omega }{\Vert }_{\infty }/P(\Omega )$, where $\Omega$ is a planar convex domain, $u_{\Omega }$ is the torsion function, $P(\Omega )$ is the perimeter of $\Omega$ and $|\Omega |$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_{\Omega }|$ attains its maximum.