JournalsifbVol. 2, No. 2pp. 201–212

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

  • Paolo Tilli

    Politecnico di Torino, Italy
On a constrained variational problem with an arbitrary number of free boundaries cover
Download PDF

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.

Cite this article

Paolo Tilli, On a constrained variational problem with an arbitrary number of free boundaries. Interfaces Free Bound. 2 (2000), no. 2, pp. 201–212

DOI 10.4171/IFB/18