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

Goldman, Michael; Ruffini, Berardo

doi:10.1515/geofl-2017-0004

Cited by 6 publications

(8 citation statements)

References 30 publications

(54 reference statements)

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“…In recent years there has been a strong interest in variational models involving a competition between a perimeter type energy and a repulsive term of long-range nature (see for instance the recent review papers [9,21] and the detailed discussion below). The aim of this paper is to start investigating the effects for this class of 132 M. Goldman, M. Novaga and M. Röger problems of higher order interfacial energies such as the Euler elastica in dimension two or the Willmore energy in dimension three.…”

Section: Introductionmentioning

confidence: 99%

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Quantitative estimates for bending energies and applications to non-local variational problems

Goldman¹,

Novaga²,

Röger³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We discuss a variational model, given by a weighted sum of perimeter, bending and Riesz interaction energies, that could be considered as a toy model for charged elastic drops. The different contributions have competing preferences for strongly localized and maximally dispersed structures. We investigate the energy landscape in dependence of the size of the 'charge', i.e. the weight of the Riesz interaction energy.In the two-dimensional case we first prove that for simply connected sets of small elastica energy, the elastica deficit controls the isoperimetric deficit. Building on this result, we show that for small charge the only minimizers of the full variational model are either balls or centered annuli. We complement these statements by a non-existence result for large charge. In three dimensions, we prove area and diameter bounds for configurations with small Willmore energy and show that balls are the unique minimizers of our variational model for sufficiently small charge. 1 We choose to keep the factor 1 4 in dimension three to stick with the traditional notation.F λ,Q (E).(1.3)We start by considering the planar problem d = 2 and first focus on the uncharged case Q = 0. For λ = 0 no global minimizer exists in M(|B 1 |), but it has been recently shown in [7,14] that balls minimize the elastica energy under volume constraint in the class of simply connected sets. Our first result is a quantitative version of this fact in the spirit of the quantitative isoperimetric inequality [18,10].Theorem 1.1. There exists a universal constant c 0 > 0 such that for every set E ∈ M sc (|B 1 |),where E∆F denotes the symmetric difference of the sets E and F . Furthermore, there exist δ 0 > 0 and c 1 > 0 such that if W (E) ≤ W (B 1 ) + δ 0 , then W (E) − W (B 1 ) ≥ c 1 (P (E) − P (B 1 )).

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Section: Introductionmentioning

confidence: 99%

“…Surprisingly, it has been shown in [24] that independently of the charge, no minimizers exist for this model (see also [41,42]). It has been suggested in [21] that a regularization by a Willmore type energy such as the one considered here might restore well-posedness. This paper can be seen as the first step in this direction.…”

Section: Introductionmentioning

confidence: 99%

Quantitative estimates for bending energies and applications to non-local variational problems

Goldman¹,

Novaga²,

Röger³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…In this paper our goal is to study the well-posedness of the problem and the regularity and formation of singularities. We refer to [45] and [29] for an introduction to the topic.…”

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confidence: 99%

“…This means that the electrostatic term is not a lower order with respect to the surface tension, which makes the problem mathematically challenging. In order to make the variational problem well-posed one may restrict the problem to convex sets [28] or regularize the functional by adding a curvature term [29] which could lead to the existence of minimizer as the result in [26] suggest.…”

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confidence: 99%

A priori estimates for the motion of charged liquid drop: A dynamic approach via free boundary Euler equations

Julin¹,

Manna²

2021

Preprint

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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 expects 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, regularity and the formation of singularities of the solution. Our main theorem roughly states that if the flow remains C 1,α -regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., C ∞ in time and space, assuming that the initial data is smooth. Due to the appearance of Taylor cones we expect the C 1,α -regularity assumption to be optimal, while the Lipschitzregularity assumption on the velocity is standard in the classical theory of the Euler equations. We also quantify the C ∞ -estimate via high order energy estimates. This result is new also for the Euler equations with free boundary without the electric field. We point out that we do not consider the problem of existence in this paper. It will be studied in forthcoming work.

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“…In spite of the interest of (1.3) in applications, a rigorous mathematical study of this model has been only performed in the last years, mostly thanks to the work of Goldman, Muratov, Novaga and Ruffini, see [15,20,17,21,16] and references therein.…”

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confidence: 99%