SYMMETRY RESULTS FOR SOME OVERDETERMINED OBSTACLE PROBLEMS

We establish symmetry results for two categories of overdetermined obstacle problems: Serrin-type problems and problems for two-phase conductors under the overdetermination that the interface serves as a level surface of the solution. The first proof avoids the method of moving planes; instead, it leverages the comparison principle and the fact that the solution of the obstacle problem is superharmonic within its domain. The second proof utilizes a suitable version of Serrin’s method of moving planes.