The Equilibrium Measure for a Nonlocal Dislocation Energy

Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia

doi:10.1002/cpa.21762

Cited by 30 publications

(54 citation statements)

References 22 publications

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“…In particular the minimality of the semi-circle law for the dislocation energy I 1 can be deduced from Theorem 1.1 by a limiting argument based on Γ-convergence (see Corollary 3.3). That is, we obtain again the main result of [29], but with a different proof based on methods from fluid mechanics and complex analysis.…”

Section: Kirchhoff Ellipses and Dislocationssupporting

confidence: 57%

“…The case α = 0 is well-known. The proof for α ∈ (0, 1) follows the lines of the analogous result for α = 1; see [29,Section 2]. For the convenience of the reader we recall the main steps of the proof.…”

Section: Existence and Uniqueness Of The Minimiser Of I αmentioning

confidence: 88%

“…In the case α = 1, the energy I 1 models interactions between edge dislocations of the same sign (see, e.g., [28,21]). The minimisers of I 1 were since long conjectured to be vertical walls of dislocations, and this has been confirmed only very recently, in [29], where the authors proved that the only minimiser of I 1 is the semi-circle law…”

Section: Introductionmentioning

confidence: 95%

“…is actually the minimiser of the energy I α and that it is possible to prove it. In fact, in the literature there are very few explicit characterisations of minimisers for nonlocal energies and the only other example in the nonradial case is the result proved in [29] corresponding to α = 1. The reason why the minimality of µ α is somewhat unexpected is the following.…”

Section: Kirchhoff Ellipses and Dislocationsmentioning

confidence: 99%

“…In fact, for α = 0, that is, for purely logarithmic interactions, it is well-known that there exists a unique minimiser of I 0 , which is given by the so-called circle law µ 0 := 1 π χ B 1 (0) (see, e.g., [20,31], and the references therein). The case α = 1, that is, the case of interacting edge dislocations, has been recently solved in [29], and it has been shown that I 1 has a unique minimiser, given by the semi-circle law (1.3). A simple comparison argument shows that µ 1 is indeed the unique minimiser of I α for any α ≥ 1.…”

Section: Existence and Uniqueness Of The Minimiser Of I αmentioning

confidence: 99%

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The Ellipse Law: Kirchhoff Meets Dislocations

Carrillo

Mateu

Mora

et al. 2019

Commun. Math. Phys.

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In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter α ∈ R. The case α = 0 corresponds to purely logarithmic interactions, minimised by the celebrated circle law for a quadratic confinement; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for α ∈ (0, 1) the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an ellipse. To prove our result we borrow techniques from fluid dynamics, in particular those related to Kirchhoff's celebrated result that domains enclosed by ellipses are rotating vortex patches, called Kirchhoff ellipses. Therefore we show a surprising connection between vortices and dislocations.

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Section: Kirchhoff Ellipses and Dislocationssupporting

confidence: 57%

Section: Existence and Uniqueness Of The Minimiser Of I αmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 95%

Section: Kirchhoff Ellipses and Dislocationsmentioning

confidence: 99%

Section: Existence and Uniqueness Of The Minimiser Of I αmentioning

confidence: 99%

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The Ellipse Law: Kirchhoff Meets Dislocations

Carrillo

Mateu

Mora

et al. 2019

Commun. Math. Phys.

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Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

Carrillo,

Shu

2023

Comm Pure Appl Math

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We study a large family of Riesz‐type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse‐supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse‐supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

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The equilibrium measure for an anisotropic nonlocal energy

et al. 2021

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In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $$I_\alpha $$ I α defined on probability measures in $${\mathbb {R}}^n$$ R n , with $$n\ge 3$$ n ≥ 3 . The energy $$I_\alpha $$ I α consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $$\alpha =0$$ α = 0 and is anisotropic otherwise, and a quadratic confinement. The two-dimensional case arises in the study of defects in metals and has been solved by the authors by means of complex-analysis techniques. We prove that for $$\alpha \in (-1, n-2]$$ α ∈ ( - 1 , n - 2 ] , the minimiser of $$I_\alpha $$ I α is unique and is the (normalised) characteristic function of a spheroid. This result is a paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. In particular, the phenomenon of loss of dimensionality, observed in dimension $$n=2$$ n = 2 , does not occur in higher dimension at the value $$\alpha =n-2$$ α = n - 2 corresponding to the sign change of the Fourier transform of the interaction potential.

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The Equilibrium Measure for a Nonlocal Dislocation Energy

Cited by 30 publications

References 22 publications

The Ellipse Law: Kirchhoff Meets Dislocations

The Ellipse Law: Kirchhoff Meets Dislocations

Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

The equilibrium measure for an anisotropic nonlocal energy

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