A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

Cicalese, Marco; Leonardi, Gian Paolo

doi:10.1007/s00205-012-0544-1

Cited by 153 publications

(184 citation statements)

References 24 publications

(48 reference statements)

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“…As it was mentioned in the introduction we will use the regularity of ƒ-minimizers to rule out those competing sets which are not regular. This idea goes back to the work by White [24] and more recently it has been used by Cicalese and Leonardi [8] and by Fusco and Morini [10]. We begin by proving a simple lemma.…”

Section: Proof Of the Main Theoremmentioning

confidence: 98%

“…in [8] where it is called strong ƒ-minimality. Very similar is the definition of 'almost minimizer' or 'quasiminimizer' of the perimeter used e.g.…”

Section: Regularity Of ƒ-Minimizersmentioning

confidence: 99%

“…The proof can be found e.g. in [8] with a few modifications due to the fact that the relative boundary is a manifold with boundary.…”

Section: Theorem 32mentioning

confidence: 99%

See 2 more Smart Citations

Minimality via second variation for microphase separation of diblock copolymer melts

Julin

Pisante

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We consider a non-local isoperimetric problem arising as the sharp interface limit of the Ohta–Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the problem. Moreover, we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the

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Section: Proof Of the Main Theoremmentioning

confidence: 98%

“…in [8] where it is called strong ƒ-minimality. Very similar is the definition of 'almost minimizer' or 'quasiminimizer' of the perimeter used e.g.…”

Section: Regularity Of ƒ-Minimizersmentioning

confidence: 99%

See 1 more Smart Citation

Minimality via second variation for microphase separation of diblock copolymer melts

Julin

Pisante

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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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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“…(A) Sharp quantitative inequalities: In [CL12], (1.1) was used (with E = B = B 0,1 ) in combination with a selection principle and a result by Fuglede on nearly spherical sets [Fug89] to give an alternative proof of the sharp quantitative isoperimetric inequality of [FMP08], namely P (E) ≥ P (B) 1 + c(n) min…”

mentioning

confidence: 99%

Improved convergence theorems for bubble clusters. I. The planar case

Cicalese¹,

Leonardi²,

Maggi³

2016

Indiana Univ. Math. J.

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Abstract. We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this construction to isoperimetric problems for planar bubble clusters. In this setting we develop an improved convergence theorem, showing that a sequence of almost-minimizing planar clusters converging in L 1 to a limit cluster has actually to converge in a strong C 1,α -sense. Applications of this improved convergence result to the classification of isoperimetric clusters and the qualitative description of perturbed isoperimetric clusters are also discussed. Analogous results for three-dimensional clusters are presented in part two, while further applications are discussed in some companion papers.

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A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

Cited by 153 publications

References 24 publications

Minimality via second variation for microphase separation of diblock copolymer melts

Minimality via second variation for microphase separation of diblock copolymer melts

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

Improved convergence theorems for bubble clusters. I. The planar case

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