A maximum-principle approach to the minimisation of a nonlocal dislocation energy

Joan Mateu, ,Lucia Scardia,Guest Editors: Lucia De Luca; Marcello Ponsiglione ,Luca Rondi,Joan Verdera,Link: <A Href="Https://Www.aimspress.com/Newsinfo/1369.Html" Target="_Blank">Https://Www.aimspress.com/Newsinfo/1369.Html</A> ,Maria Giovanna Mora

doi:10.3934/mine.2020012

Abstract

In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies Iα defined on probability measures in $\mathbb{R}^2$. The purely nonlocal term in Iα is of convolution type, and is isotropic for α = 0 and anisotropic otherwise. The cases α = 0 and α = 1 are special: The first corresponds to Coulombic interactions, and the latter to dislocations. The minimisers of Iα have been characterised by the same authors in an earlier paper, by exploiting some formal similarities with the Euler equation, and by means of complex-analysis techniques. We here propose a different approach, that we believe can be applied to more general energies.

Highlights

We consider the family of nonlocal energies Iα(μ) =Wα(x − y) dμ(x) dμ(y) + |x|2 dμ(x), R2×R2 R2 (1.1)defined on probability measures μ ∈ P(R2), where the interaction potential Wα is given by Wα(x) = W0(x) + α x12 |x|2
In this paper we use an approach based on the maximum principle to characterise the minimiser of a family of nonlocal and anisotropic energies Iα defined on probability measures in R2
The purely nonlocal term in Iα is of convolution type, and is isotropic for α = 0 and anisotropic otherwise

Summary

Introduction

Defined on probability measures μ ∈ P(R2), where the interaction potential Wα is given by. The minimiser of Iα is the (normalised) characteristic function of an ellipse for α ∈ (−1, 1), and it converges to a singular, one-dimensional measure (the semi-circle law) for α → ±1 This result has been proved in [7] by means of complex-analysis techniques, and in [16] via a more direct proof, based on the explicit computation of the potential Wα ∗ μα in R2. The idea is to construct an auxiliary function gα, harmonic outside Ωα, and to do so in such a careful and clever way that the application of the standard maximum principle for harmonic functions to gα gives, as a welcomed byproduct, the unilateral condition (1.5) for fα. The idea for this construction is taken from the work [9], where the author formulates several variants of the maximum principle that are valid for biharmonic functions

Characterisation of the minimiser of Iα via the maximum principle

The potential inside an ellipse

Conflict of interest