Hessian degeneracy of the torsion function on smooth simply connected nonconvex planar domains
We construct a family of bounded, smooth, simply connected, nonconvex planar domains $\Omega_{a,\epsilon}$. Let $u_{a,\epsilon}$ be the corresponding torsion function satisfying \[ -\Delta u_{a,\epsilon}=1\quad\text{in }\Omega_{a,\epsilon},\qquad u_{a,\epsilon}=0\quad\text{on }\partial\Omega_{a,\epsilon}. \] There exists \(a^*\in(0,1)\) such that, for every \(a\in(a^*,1)\) and all sufficiently small \(\epsilon>0\), the origin is a unique global maximizer of $u_{a,\epsilon}$. Moreover, \[ \lim_{a\downarrow a^*}\lim_{\epsilon\to0} \lambda_{\max}\bigl(D^2u_{a,\epsilon}(0)\bigr)=0. \] In addition, the ratios $\text{diam}(\Omega_{a,\epsilon}) / \text{inrad}(\Omega_{a,\epsilon})$ are uniformly bounded. Hence the Hessian estimate proved by Steinerberger (J. Funct. Anal. 274, 1611--1630, 2018) for convex planar domains cannot be extended to smooth, simply connected, nonconvex planar domains. \vskip0.2cm Our domains are star-shaped, symmetric with respect to both coordinate axes and convex in the horizontal direction. When the limiting slit is sufficiently long, we further show that certain superlevel sets of the torsion function are not star-shaped. This strengthens the counterexample to the star-shapedness question raised by Gladiali and Grossi (Amer. J. Math. 144, 1221--1240, 2022) under stronger geometric assumptions.