On a Volume-Constrained Variational Problem

Ambrosio, Luigi; Fonseca, Irene; Marcellini, Paolo; Tartar, Luc

doi:10.1007/s002050050166

Cited by 22 publications

(50 citation statements)

References 13 publications

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“…Now we establish some properties of the minimizers of F λ , which generalize those obtained in [4] for the case of two levels. THEOREM 2.2 If u ∈ H 1 (Ω) is a minimum for F λ and (13) holds, then…”

mentioning

confidence: 73%

“…In addition, several properties of minimizers where obtained in [4]; in particular, the asymptotic behavior of solutions was identified, when the total volume of the different phases exhaust the total measure of Ω.…”

Section: P Tillimentioning

confidence: 99%

“…u is real valued), and our goal is twofold: on one hand, we prove the existence of minimizers of (2) for an arbitrary number of levels; on the other hand, we study their regularity, which was not investigated in [4].…”

Section: P Tillimentioning

confidence: 99%

“…, m (observe that no boundary condition is given). The above problem was first introduced by Ambrosio et al in [4], motivated by a question posed by Gurtin in connection with a problem on the interface between immiscible fluids.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], the authors established existence of minimizers under certain assumptions on the vectors l i , namely that they should be extreme points of a convex set of R n ; in the scalar case this condition reduces to considering only two phases, i.e. m = 1.…”

Section: Introductionmentioning

confidence: 99%

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

Tilli

2000

Interfaces Free Bound.

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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.

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confidence: 73%

Section: P Tillimentioning

confidence: 99%

Section: P Tillimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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