  We present a quantitative estimate for the radially symmetric configuration
concerning a Serrin-type overdetermined problem for the torsional rigidity in a
bounded domain $\Omega $, when the equation is known on $\Omega \setminus
\overline{\omega}$ only, for some open subset $\omega \Subset \Omega$. The
problem has concrete motivations in optimal heating with malfunctioning,
laminar flows and beams with small inhomogeneities.
