The simultaneous asymmetric perturbation method for overdetermined free boundary problems

In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin's overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.