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We provide a full characterization of multi-phase problems under a large class of overdetermined Serrin-type conditions. Our analysis includes both symmetry and asymmetry (including bifurcation) results. A broad range of techniques is needed to obtain a full characterization of all the cases, including applications of results obtained via the moving planes method, approaches via integral identities in the wake of Weinberger, applications of the Crandall-Rabinowitz theorem, and the Chauchy-Kovalevskaya theorem. The multi-phase setting entails intrinsic difficulties that make it difficult to predict whether a given overdetermination will lead to symmetry or asymmetry results; the results of our analysis are significant as they answer such a question providing a full characterization of both symmetry and asymmetry results., Comment: 39 pages, 2 figures

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., Comment: 23 pages, 5 figures