The spectral gap to torsion problem for some non-convex domains

In this paper we study the following torsion problem \begin{equation*} \begin{cases} -\Delta u=1~&\mbox{in}\ \Omega,\\[1mm] u=0~&\mbox{on}\ \partial\Omega. \end{cases} \end{equation*} Let $\Omega\subset \mathbb{R}^2$ be a bounded, convex domain and $u_0(x)$ be the solution of above problem with its maximum $y_0\in \Omega$. Steinerberger proved that there are universal constants $c_1, c_2>0$ satisfying \begin{equation*} \lambda_{\max}\left(D^2u_0(y_0)\right)\leq -c_1\mbox{exp}\left(-c_2\frac{\text{diam}(\Omega)}{\mbox{inrad}(\Omega)}\right). \end{equation*} And he proposed following open problem: "Does above result hold true on domains that are not convex but merely simply connected or perhaps only bounded? The proof uses convexity of the domain $\Omega$ in a very essential way and it is not clear to us whether the statement remains valid in other settings." Here by some new idea involving the computations on Green's function, we compute the spectral gap $\lambda_{\max}D^2u(y_0)$ for some non-convex smooth bounded domains, which gives a negative answer to above open problem. Also some extensions are given.