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In a paper Alessandrini and Isakov consider the inverse problem of determining the coefficient of an elliptic equation of divergence form in a bounded smooth domain with over-determined data assigned on the boundary. In order to show uniqueness for the conductivity coefficient the key point leads to a free boundary problem. They show that the free boundary is analytic surface under the assumption of Holder continuous normal. In a joint work with L.A. Caffarelli and S. Salsa we study under which weaker hypotheses on this problem can we still assert the free boundary is smooth. This would enlarge the family of configurations for which the inverse is well posed. Two natural hypotheses are a) free boundary is Lipschitz b) free boundary is a set of finite perimeter. Some transversality condition is also necessary as pointed out in their paper.
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