Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

Figalli, Alessio; Fusco, Nicola; Maggi, Francesco; Millot, Vincent; Morini, Massimiliano

doi:10.1007/s00220-014-2244-1

Cited by 170 publications

(264 citation statements)

References 32 publications

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“…Furthermore, the physical picture of identical radial droplets in the limit is expected for sufficiently long-ranged kernels, i.e., those kernels that satisfy G(x, y) ∼ |x − y| −α for |x − y| 1, with 0 < α 1 [36,47]. Note that although a similar characterization of the minimizers for long-ranged kernels exists in higher dimensions as well [44,46], these results are still not sufficient to be used to characterize the limit droplets, since they do not give an explicit interval of existence of the minimizers of the self-interaction problem. Also, since the non-existence result for the self-energy with such kernels is available only for α < 2 [37], our results may not extend to the case of α ≥ 2 in dimensions three and above.…”

Section: -3mentioning

confidence: 91%

“…Importantly, the model in (1.1) is a paradigm for the energy-driven pattern forming systems in which spatial patterns (global or local energy minimizers) form as a result of the competition of short-range attractive and long-range repulsive forces. This is why this model and its generalizations attracted considerable attention of mathematicians in recent years (see, e.g., [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], this list is certainly not exhaustive). In particular, the volume-constrained global minimization problem for (1.1) in the whole space with no neutralizing background, which we will also refer to as the "self-energy problem", has been investigated in [34,37,45,49].…”

Section: -3mentioning

confidence: 99%

“…Furthermore, for sufficiently small δ we have H ρ ≥ 0 and 45) for some universal c > 0 (where Γ is the Newtonian potential in R 3 , as above). From (5.42), (5.44) and (5.45) we then get 46) for ε small enough. Letting nowû ε (x) :=ũ ε (ε 1/3 x) be the rescaled function which satisfies 47) for every fixed ρ 0 > 0 and ε sufficiently small, we get 48) where Γ ρ 0 is defined via (5.21).…”

Section: Equidistribution Of Energymentioning

confidence: 99%

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Low Density Phases in a Uniformly Charged Liquid

Knüpfer

Muratov

Novaga

2016

Commun. Math. Phys.

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This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta-Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase.Under periodic boundary conditions, we prove that the considered energy Γ-converges to an energy functional of the limit "homogenized" measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplets per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

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Section: -3mentioning

confidence: 91%

Section: -3mentioning

confidence: 99%

Section: Equidistribution Of Energymentioning

confidence: 99%

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Low Density Phases in a Uniformly Charged Liquid

Knüpfer

Muratov

Novaga

2016

Commun. Math. Phys.

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“…6.4] for the logarithmic case when N = ) that for small enough charges, the ball is the unique minimizer of (2.2). The proof of this result is quite long and involved but the basic idea is to argue as in [5,9,18,22] for instance and show that for small charges minimizers are nearly spherical sets, that is small Lipschitz graphs over ∂B . This allows the use of a Taylor expansion of the perimeter for this type of sets given by Fuglede [13].…”

Section: R > There Exists a Charge Q(r) > Such That The Ball Of Radiumentioning

confidence: 99%

Equilibrium shapes of charged droplets and related problems: (mostly) a review

Goldman¹,

Ruffini²

2017

Geometric Flows

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Abstract:We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric eld. Second, we prove that there exists no optimal conducting drop in this setting.Keywords: Isoperimetry, non-local energies, charged dropsThe main purpose of the paper is to review some recent progress in the study of variational problems describing the shape of conducting liquid drops. The salient feature of these models is the competition between an interfacial term with a non-local and repulsive term of capacitary type. The somewhat surprising and puzzling fact is that contrarily to the experimental observations these models are generally ill-posed. By this we mean that they never admit minimizers. However, taking into account various possible regularizing mechanisms, it is possible in some cases to recover well-posedness together with stability results for the ball in the regime of small charges. Alongside this review, we also provide new results on the closely related problem of equilibrium shapes of conducting drops subject to an external electric eld. We prove that for every xed drop, the optimal distribution of charges exists but that similarly to the charged drops model, no equilibrium shape exists. Moreover, we show that in contrast with the charged drop model, non-existence still holds in the rigid class of convex sets.The paper is organized as follows. In Section 1 we recall the de nition of the capacity and study existence and characterization of optimal distributions of charges. In Section 2 we review some recent results on the charged liquid drop model. In particular, we show ill-posedness of this problem and discuss various possible regularizations. In Section 3 we study the problem of equilibrium shapes for perfectly conducting drops in an external electric eld. In the last section, we state several open problems.

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“…[4,1,11] for further details on this notion of mean curvature. Notice that, if E is a critical point of F ε with boundary of class C 2 , by elliptic regularity (see [2]) the boundary of E is in fact of class C ∞ .…”

Section: Introductionmentioning

confidence: 99%

Rigidity of critical points for a nonlocal Ohta–Kawasaki energy

2017

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We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers.We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.

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Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

Cited by 170 publications

References 32 publications

Low Density Phases in a Uniformly Charged Liquid

Low Density Phases in a Uniformly Charged Liquid

Equilibrium shapes of charged droplets and related problems: (mostly) a review

Rigidity of critical points for a nonlocal Ohta–Kawasaki energy

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