Concavity of the Ornstein--Uhlenbeck torsion function on centered ellipses and its failure on convex domains

Let $u$ be the torsion function of a bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$ for the Ornstein–Uhlenbeck operator, that is, the solution of $-\mathrm{\Delta }u+x\cdot \mathrm{\nabla }u=1$ in $\mathrm{\Omega }$ with $u=0$ on the boundary. We prove that for every $n\ge 2$ there are bounded, centrally symmetric domains whose torsion function is not concave. These domains have boundary with a flat piece and can be smoothed so that the boundary is $C^{\mathrm{\infty }}$, strictly convex with principal curvatures bounded below by a positive constant, while keeping the torsion function not concave. In contrast, the torsion function for every ellipse centered at the origin is concave, and for every ball centered at the origin we prove the sharp bound $D^{2}u\le -\frac{1}{n}I$. The ellipse result follows by deformation from the disk, using a boundary flux estimate and third-order boundary identities to exclude a first zero eigenvalue of the Hessian.