On a constrained variational problem with an arbitrary number of free boundaries

Abstract

We study the problem of minimizing the Dirichlet integral among all functions u [isin] H1([ohm]) whose level sets {u=li} have prescribed Lebesgue measure [agr]i. This problem was introduced in connection with a model for the interface between immiscible fluids. The existence of minimizers is proved with an arbitrary number of level-set constraints, and their regularity is investigated. Our technique consists in enlarging the class of admissible functions to the whole space H1([ohm]), penalizing those functions whose level sets have measures far from those required; in fact, we study the minimizers of a family of penalized functionals F[lgr], [lgr] > 0, showing that they are Höder continuous, and then we prove that such functions minimize the original functional also, provided the penalization parameter [lgr] is large enough. In the case where only two levels are involved, we prove Lipschitz continuity of the minimizers.