arXiv subjects

L. Scardia

Publications and source records attributed to L. Scardia.

3 recordsLinked to original sources

arXiv2020

Explicit minimisers of some nonlocal anisotropic energies: a short proof

In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+α\, x^2/|z|^2, \; z=x+iy,$ with $-1 < α< 1.$ This kernel is anisotropic except for the Coulombic case $α=0.$ We present a short compact proof of the known surprising fact that the unique minimiser of the energy is the normalised characteristic function of the domain enclosed by an ellipse with horizontal semi-axis $\sqrt{1-α}$ and vertical semi-axis $\sqrt{1+α}.$ Letting $α\to 1^-$ we find that the semicircle law on the vertical axis is the unique minimiser of the corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible

math.CA↗

arXiv2019

The equilibrium measure for an anisotropic nonlocal energy

In this paper we characterise the minimisers of a one-parameter family of nonlocal and anisotropic energies $I_α$ defined on probability measures in $\R^n$, with $n\geq 3$. The energy $I_α$ consists of a purely nonlocal term of convolution type, whose interaction kernel reduces to the Coulomb potential for $α=0$ 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 $α\in (-1, n-2]$, the minimiser of $I_α$ 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 $n=2$, does not occur in higher dimension at the value $α=n-2$ corresponding to the sign change of the Fourier transform of the interaction potential.

math.AP↗

arXiv2017

The ellipse law: Kirchhoff meets dislocations

In this paper we consider a nonlocal energy $I_α$ whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter $α\in \R$. 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 $α\in (0,1)$ the minimiser can be computed explicitly and is the normalised characteristic function of the domain enclosed by an \emph{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 \emph{Kirchhoff ellipses}. Therefore we show a surprising connection between vortices and dislocations.

math.AP↗