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

*(English)*Zbl 0995.49002Summary: 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 HĂ¶lder 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.

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 HĂ¶lder 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 |