A Serrin-type problem with partial knowledge of the domain

Abstract

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 $\omega ⋐\Omega$. The problem has concrete motivations in optimal heating with malfunctioning, laminar flows and beams with small inhomogeneities.