A Serrin-type problem with partial knowledge of the domain

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