Geodesic
On a constrained variational problem with an arbitrary number of free boundaries
Paolo Tilli1
1Politecnico di Torino, Italy
Interfaces and free boundaries, Tome 2 (2000) no. 2, pp. 201-212

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DOI

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.
DOI : 10.4171/ifb/18
Classification : 46-XX, 60-XX
Mots-clés : constrained variational problems; free boundary problems; immiscible fluids

Paolo Tilli  1

1 Politecnico di Torino, Italy 
Paolo Tilli. On a constrained variational problem with an arbitrary number of free boundaries. Interfaces and free boundaries, Tome 2 (2000) no. 2, pp. 201-212. doi: 10.4171/ifb/18
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