Low energy configurations of topological singularities in two dimensions: A $\Gamma$-convergence analysis of dipoles

created by ponsiglio on 12 Nov 2017
modified by deluca on 13 Sep 2021

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Published Paper

Inserted: 12 nov 2017
Last Updated: 13 sep 2021

Journal: Comm. Contemp. Math.
Volume: 22
Number: 3
Year: 2020

Abstract:

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the {core radius approach} and the Ginzburg-Landau energy. Denoting by $\varepsilon$ the length scale parameter in such models, we focus on the $\log\frac{1}{\varepsilon}$ energy regime. It is well known that, for configurations whose energy is bounded by $c \log \frac{1}{\varepsilon}$, the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying $\pi \log \frac{1}{\varepsilon}$ energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses.

Here we perform a compactness and $\Gamma$-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale $\varepsilon^s$, for $0<s<1$), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical $\Gamma$-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order $c \log \frac{1}{\varepsilon}$ with $c<\pi$.