A counterexample with convex domain to a conjecture of De Saint Venant (Q908733)

The elliptic problem (1) \(-\Delta u=1\) in \(\Omega \subset {\mathbb{R}}^ 2\), \(u=0\) on \(\partial \Omega\), in \(\Omega\) which has the orthogonal axes as axes of symmetry, was first studied by B. De Saint Venant. The function \(\nabla u\) contains the stress components of an elastic bar, with cross section \(\Omega\), under torsion. The points of interest for mechanical engineers are the points where the stress, \(| \nabla u(x)|\), becomes maximal. It is still believed by some that the following conjecture holds: For convex domains \(\Omega\), \(| \nabla u(x)|\) attains its maximum on the intersection of \(\partial \Omega\) and the largest inscribed circle. We give an example where this conjecture leads to a contradiction. Hence we may state: The conjecture is not true in general.