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 ∈ H 1 (Ω) whose level sets {u = l i } have prescribed Lebesgue measure α 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 (Ω), 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 λ , λ > 0, showing that they are Hölder continuous, and then we prove that such functions minimize the original functional also, provided the penalization parameter λ is large enough. In the case where only two levels are involved, we prove Lipschitz continuity of the minimizers.

On a variational problem with non-differentiable constraints

“…This is true, for instance, for the problem given by (5), as the next proposition shows. This result can be proved by arguments that were already used in [13] for a similar statement. For completeness, we include a proof in the appendix.…”

“…Thus they solve even the original problem. This situation has first been studied by Tilli [13] in the form considered here, but similar problems have been solved earlier by Aguilera et al [1] (using tools from Alt and Caffarelli [2]) and by Ambrosio et al [3]. Other related works are by Mosconi and Tilli [8], Stepanov and Tilli [12], Morini and Rieger [7], Rieger [11], and Leonardi and Tilli [5].…”

“…Other related works are by Mosconi and Tilli [8], Stepanov and Tilli [12], Morini and Rieger [7], Rieger [11], and Leonardi and Tilli [5]. It is proved in [13] that E has a minimizer under the constraint (2) if f is of the form (5), and moreover, minimizers are Hölder continuous. This is achieved by replacing the constraint with a penalization term added to E.…”

On a Volume Constrained Variational Problem in SBV²(Ω): Part I

“…In our case we were unable to drop this assumption and that is why we only consider two level sets. We remark, however, that under the hypothesis m = 1 it was established by Tilli [12] that minimizers of I(·) are, in fact, locally Lipschitz continuous.…”

“…We refer also to [3] and [4] where related problems were treated. A further step was taken by Tilli [12], in the scalar case, who established locally Hölder continuity of minimizers of I(·) and was able to drop the extremality assumption needed in [2] which, in the scalar case, is equivalent to the restriction m = 1, i.e. only two level sets are allowed.…”