The Brunn-Minkowski inequality for p -capacity of convex bodies

Colesanti, Andrea; Salani, Paolo

doi:10.1007/s00208-003-0460-7

Cited by 84 publications

(97 citation statements)

References 34 publications

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“…As mentioned in the introduction, a proof of the existence and uniqueness of U can be found in Lewis [57], see also Theorem 2 in [28]. For the following theorem we refer to Lewis [57].…”

Section: Basics Of Convex Domains By Definition a Domain In Rmentioning

confidence: 99%

“…Our proof is quite different compared to the proof of Jerison, although it follows the same general scheme, and it relies on the Brunn-Minkowski inequality for p-capacity established by Colesanti and Salani, see [28]. We use the Hadamard variational formula (1.13) and the Colesanti-Salani Brunn-Minkowski inequality to establish the following uniqueness result for the Minkowski problem for pcapacity.…”

mentioning

confidence: 99%

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The Hadamard variational formula and the Minkowski problem for p-capacity

Colesanti

Nyström

Salani

et al. 2015

Advances in Mathematics

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“…As mentioned in the introduction, a proof of the existence and uniqueness of U can be found in Lewis [57], see also Theorem 2 in [28]. For the following theorem we refer to Lewis [57].…”

Section: Basics Of Convex Domains By Definition a Domain In Rmentioning

confidence: 99%

mentioning

confidence: 99%

The Hadamard variational formula and the Minkowski problem for p-capacity

Colesanti

Nyström

Salani

et al. 2015

Advances in Mathematics

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“…We prove that the reverse inequality holds in (5) and (6). More precisely, we shall prove that if u is a solution of (3), then it satisfies…”

Section: Lemma 22 For Any Smooth Function V and γ ∈ R We Have The Fmentioning

confidence: 88%

A Note on an Overdetermined Problem for the Capacitary Potential

Bianchini

Ciraolo

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…The characterization at in nity of the behavior of the potential described in point ( ) of the previous theorem has been successfully exploited to show geometric inequalities, such as Brunn-Minkowskitype inequalities [2,3,6,29].…”

Section: Remark 13mentioning

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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The Brunn-Minkowski inequality for p -capacity of convex bodies

Cited by 84 publications

References 34 publications

The Hadamard variational formula and the Minkowski problem for p-capacity

The Hadamard variational formula and the Minkowski problem for p-capacity

A Note on an Overdetermined Problem for the Capacitary Potential

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

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