On convex domains maximizing the gradient of the torsion function

We consider the solution of $-\mathrm{\Delta }u=1$ on convex domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$ subject to Dirichlet boundary conditions $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$. Our main concern is the behavior of $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$, also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\mathrm{/}\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{1\mathrm{/}2}$ and $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\mathrm{/}{H}^1(\mathrm{\partial }\mathrm{\Omega })$. Numerically, the extremal domain for each functional looks a bit like the rounded letter `D’. We prove that (1) either the extremal domain does not have a $C^{2+\epsilon }$ boundary or (2) there exists an infinite set of points on $\mathrm{\partial }\mathrm{\Omega }$ where the curvature vanishes. Either scenario seems curious and is rarely encountered for such problems. The techniques are based on finding a representation of the functional using only conformal geometry and classic perturbation arguments.