Face 2-phase: how much overdetermination is enough to get symmetry in two-phase problems

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.