The Ellipse Law: Kirchhoff Meets Dislocations

Carrillo, José A.; Mateu, Jоan; Mora, Maria Giovanna; Rondi, Luca; Scardia, Lucia; Verdera, Jоan

doi:10.1007/s00220-019-03368-w

Cited by 28 publications

(69 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our result generalises to any dimension n ≥ 3 the work [10] and is another paradigmatic example of the role of the anisotropy of the kernel on the shape of minimisers. Our approach here is however completely different from the approach in [10], which was based on complexanalysis techniques, clearly no longer available in higher dimensions.…”

Section: Our Approach and Discussionsupporting

confidence: 59%

“…Explicitly characterising the minimiser, or even understanding its shape and general properties, is therefore much more challenging in the case of anisotropic interactions. To the best of our knowledge, this has been previously done only in [10,25], in dimension n = 2.…”

Section: Our Approach and Discussionmentioning

confidence: 99%

“…A key step in [10] was to compute exactly the (gradient of the) potential W α * μ a,b , with μ a,b being the (normalised) characteristic function of an ellipse of semi-axes a and b, and to impose the Euler-Lagrange conditions associated to I α . Also for n ≥ 3 we prove the minimality of spheroids via the Euler-Lagrange conditions [see (3.1)…”

Section: Our Approach and Discussionmentioning

confidence: 99%

“…Note that W 1 = W edge for n = 2. The two-dimensional case was considered in [10,25] for every α ∈ R.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The equilibrium measure for an anisotropic nonlocal energy

et al. 2021

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Our Approach and Discussionsupporting

confidence: 59%

Section: Our Approach and Discussionmentioning

confidence: 99%

Section: Our Approach and Discussionmentioning

confidence: 99%

“…Note that W 1 = W edge for n = 2. The two-dimensional case was considered in [10,25] for every α ∈ R.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The equilibrium measure for an anisotropic nonlocal energy

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…To the best of our knowledge, there are only few results available so far for non-symmetric potentials. Examples are [14] and [7]. In this paper, we analyse minimisers for not necessarily symmetric attractive-repulsive potentials and consider the following energy functional…”

Section: Introductionmentioning

confidence: 99%

Global minimisers for anisotropic attractive–repulsive interactions

Kaib¹,

Kang

Stevens

2019

Eur. J. Appl. Math

View full text Add to dashboard Cite

We prove the existence of global minimisers for a class of attractive–repulsive interaction potentials that are in general not radially symmetric. The global minimisers have compact support. For potentials including degenerate power-law diffusion, the interaction potential can be unbounded from below. Further, a formal calculation indicates that for non-symmetric potentials global minimisers may neither be radial symmetric nor unique.

show abstract

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

Carrillo,

Shu

2023

Comm Pure Appl Math

View full text Add to dashboard Cite

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.

show abstract

The Ellipse Law: Kirchhoff Meets Dislocations

Cited by 28 publications

References 42 publications

The equilibrium measure for an anisotropic nonlocal energy

The equilibrium measure for an anisotropic nonlocal energy

Global minimisers for anisotropic attractive–repulsive interactions

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

Contact Info

Product

Resources

About