  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.
