  In this paper, we deal with an overdetermined problem of Serrin-type with
respect to a two-phase elliptic operator in divergence form with piecewise
constant coefficients. In particular, we consider the case where the two-phase
overdetermined problem is close to the one-phase setting. First, we show
quantitative stability estimates for the two-phase problem via a one-phase
stability result. Furthermore, we prove non-existence for the corresponding
inner problem by the aforementioned two-phase stability result.
