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arXiv · 2606.12644

The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~{II}: Refined structure of the energy-concentration set

Giacomo Canevari·Federico Luigi Dipasquale·Bianca Stroffolini
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Abstract

In this paper, we continue our study, started in~\cite{CDS1}, of a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\Q$-tensor for the liquid crystal component and a magnetisation vector field~$\M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. While in~\cite{CDS1} we showed that the (rescaled) energy density for the~$\Q$-component concentrates, to leading order, on a finite number of singular points, in this paper we prove the energy density for the~$\M$-component concentrates along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component. Crucial to our arguments will be the energy estimates and compactness results proved in~\cite{CDS1}.

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Giacomo Canevari, Federico Luigi Dipasquale, Bianca Stroffolini. 2026-06-10. The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~{II}: Refined structure of the energy-concentration set. https://arxiv.org/abs/2606.12644

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