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In this paper, we introduce the notion of variational free boundary problem.Namely, we say that a free boundary problem is variational if its solutions canbe characterized as the critical points of some shape functional. Moreover, weextend the notion of nondegeneracy of a critical point to this setting. As aresult, we provide a unified functional-analytical framework that allows us toconstruct families of solutions to variational free boundary problems wheneverthe shape functional is nondegenerate at some given solution. As a clarifyingexample, we apply this machinery to construct families of nontrivial solutionsto the two-phase Serrin's overdetermined problem in both the degenerate andnondegenerate case.

How to construct parametrized families of free boundaries near nondegenerate solutions