  We consider the solution of the torsion problem $-\Delta u=1$ in $\Omega$ and
$u=0$ on $\partial \Omega$. Serrin's celebrated symmetry theorem states that,
if the normal derivative $u_\nu$ is constant on $\partial \Omega$, then
$\Omega$ must be a ball. In a recent paper, it has been conjectured that
Serrin's theorem may be obtained {\it by stability} in the following way:
first, for the solution $u$ of the torsion problem prove the estimate $$
r_e-r_i\leq C_t\,\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr) $$ for some
constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an
annulus containing $\partial\Omega$ and $\Gamma_t$ is a surface parallel to
$\partial\Omega$ at distance $t$ and sufficiently close to $\partial\Omega$;
secondly, if in addition $u_\nu$ is constant on $\partial\Omega$, show that $$
\max_{\Gamma_t} u-\min_{\Gamma_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. $$ In this
paper, we analyse a simple case study and show that the scheme is successful if
the admissible domains $\Omega$ are ellipses.
