Existence and Stability for a Non-Local Isoperimetric Model of Charged Liquid Drops

Goldman, Michael; Novaga, Matteo; Ruffini, Berardo

doi:10.1007/s00205-014-0827-9

Cited by 37 publications

(71 citation statements)

References 41 publications

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“…Roughly speaking this is due to the fact that the perimeter term sees objects of dimension N − while Iα naturally lives on object of dimension N − α, see Proposition 1.6. The following non-existence result has been obtained in [19,Th. 3.2].…”

Section: Equilibrium Shapes Of Charged Liquid Dropsmentioning

confidence: 89%

“…Most of the results can be found in [19,20,23]. For xed α ∈ ( , N), and Radon measures µ, ρ we de ne where the class of minimization runs over all probability measures on Ω.…”

Section: Equilibrium Measures Potentials and Capacitiesmentioning

confidence: 99%

“…3.2]. Performing a more careful analysis it can be shown that actually even local minimizers do not exist for the Hausdor topology (see [19,Th. 3.4]).…”

Section: Equilibrium Shapes Of Charged Liquid Dropsmentioning

confidence: 99%

“…A rst possibility, explored in [19] is to add a strong constraint on the curvature. For δ > , we say that a set Ω satis es the δ−ball condition if for every x ∈ ∂Ω there are two balls of radius δ touching at x, one of which is contained in Ω and the other one which is contained in Ω c .…”

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

confidence: 99%

“…5.6] (see also [19,Cor. 6.4] for the logarithmic case when N = ) that for small enough charges, the ball is the unique minimizer of (2.2).…”

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

confidence: 99%

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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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Section: Equilibrium Shapes Of Charged Liquid Dropsmentioning

confidence: 89%

“…Most of the results can be found in [19,20,23]. For xed α ∈ ( , N), and Radon measures µ, ρ we de ne where the class of minimization runs over all probability measures on Ω.…”

Section: Equilibrium Measures Potentials and Capacitiesmentioning

confidence: 99%

“…3.2]. Performing a more careful analysis it can be shown that actually even local minimizers do not exist for the Hausdor topology (see [19,Th. 3.4]).…”

Section: Equilibrium Shapes Of Charged Liquid Dropsmentioning

confidence: 99%

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

confidence: 99%

“…5.6] (see also [19,Cor. 6.4] for the logarithmic case when N = ) that for small enough charges, the ball is the unique minimizer of (2.2).…”

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

confidence: 99%

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

Goldman¹,

Ruffini²

2017

Geometric Flows

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On Equilibrium Shape of Charged Flat Drops

Muratov

Novaga

Ruffini

2018

Comm Pure Appl Math

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Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.

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A Priori Estimates for the Motion of Charged Liquid Drop: A Dynamic Approach via Free Boundary Euler Equations

Julin,

La Manna

2024

J. Math. Fluid Mech.

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We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains $$C^{1,\alpha }$$ C 1 , α -regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., $$C^\infty $$ C ∞ in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the $$C^{1,\alpha }$$ C 1 , α -regularity assumption to be optimal. We also quantify the $$C^\infty $$ C ∞ -regularity via high order energy estimates which, in particular, implies the well-posedness of the problem.

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Existence and Stability for a Non-Local Isoperimetric Model of Charged Liquid Drops

Cited by 37 publications

References 41 publications

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

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

On Equilibrium Shape of Charged Flat Drops

A Priori Estimates for the Motion of Charged Liquid Drop: A Dynamic Approach via Free Boundary Euler Equations

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