In this talk, we consider the mathematical model of a two-phase composite medium whose interfaces exhibit imperfect contact due to corrosion. In this setting, we study an overdetermined problem of Serrin-type, that is an elliptic linear PDE where both Dirichlet and Neumann boundary conditions are imposed at the same time. We remark that, since the solutions must simultaneously satisfy two boundary conditions, the solvability of such an overdetermined problem is deeply linked to the geometry of the composite medium (that is, the geometric shape of both the exterior boundary and the interface). An elementary example of a configuration where the overdetermined problem is solvable is the one where the exterior boundary and the interface are concentric spheres (the so-called "trivial solution").
This talk aims to study the geometry of nontrivial configurations where the overdetermined problem can be solved. We give a complete characterization of such configurations in a neighborhood of the trivial one. In particular, we show how the degeneracy of some Lagrangian is related to different symmetry behaviors of the solutions.
This talk is based on a joint work with Toshiaki Yachimura (Kyoto University).

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