Abstract
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 {alpha in mathbb{R}}. The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that for {alpha in (0, 1)} the minimiser is the normalised characteristic function of the domain enclosed by the ellipse of semi-axes {sqrt{1-alpha}} and {sqrt{1+alpha}}. This result is one of the very few examples where the minimiser of a nonlocal anisotropic energy is explicitly computed. For the proof 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.
Highlights
The starting point of our analysis is the nonlocal energy Iα (μ) = R2 ×R2 Wα (x − y) dμ(x) dμ(y) + |x|2 dμ(x) R2 (1.1)
For α ∈ (−1, 1) we prove that the unique minimiser of Iα is the n√ormalised ch√aracteristic function of the region surrounded by an ellipse of semi-axes 1 − α and 1 + α
For α = 0, that is, for purely logarithmic interactions, it is well-known that there exists a unique minimiser of
Summary
Defined on probability measures μ ∈ P(R2), where the interaction potential Wα is given by. The classical case corresponding to α = 0 in (1.2), that is, the repulsive Newtonian interaction with quadratic confinement, was analysed in [6] They showed that all the solutions of the corresponding gradient flow equation (1.9) converge as t → ∞ to the suitably normalised characteristic function of an Euclidean ball with a certain radius. Their results imply a dynamic proof of the classical minimisation result of Frostman [26,39].
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