  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 that would have been out of reach
otherwise. Another benefit of this method lies in the improvement of the
regularity gap between the free boundary and the given one. Finally, some
geometric properties of the solutions, such as symmetry and convexity, are also
discussed.
