Shape optimisation of the maximum of the gradient of the torsion function on convex sets

Abstract

We analyse the boundary behaviour of the gradient of the torsion function $w_\Om$ in convex domains $\Om\subset\mathbb{R}^d$. We prove that its $L^\infty$-norm is attained at a boundary point and establish the stability of this norm under general (convex) perturbations. For problems of the form $$\sup\{\|\nabla w_{\Om}\|_\infty: \Om \subset \R^d,\textup{open, bounded and convex}, G(\Om)=m\},$$ where $G$ is either volume or perimeter, we show the existence of maximisers which are $C^1$-smooth, rotationally symmetric about an axis and contain a $(d-1)$- dimensional ball (of estimated size) in their boundary. Other constraints, whether geometric (such as diameter or circumradius) or variational (such as torsion or first Dirichlet eigenvalue), are also briefly discussed, and some of their characteristic behaviours are highlighted.