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On a constrained variational problem with an arbitrary number of free boundaries. (English) Zbl 0995.49002
Summary: We study the problem of minimizing the Dirichlet integral among all functions \(u\in H^1(\Omega)\) whose level sets \(\{u= l_i\}\) have prescribed Lebesgue measure \(\alpha_i\). 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 \(H^1(\Omega)\), penalizing those functions whose level sets have measures far from those required; in fact, we study the minimizers of a family of penalized functionals \(F_\lambda\), \(\lambda> 0\), showing that they are Hölder continuous, and then we prove that such functions minimize the original functional also, provided the penalization parameter \(\lambda\) is large enough. In the case where only two levels are involved, we prove Lipschitz continuity of the minimizers.

MSC:
49J10 Existence theories for free problems in two or more independent variables
35R35 Free boundary problems for PDEs
35A15 Variational methods applied to PDEs
35A20 Analyticity in context of PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control
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