arXiv subjects

Federico Luigi Dipasquale

Publications and source records attributed to Federico Luigi Dipasquale.

8 recordsLinked to original sources

arXiv2026

Ferronematics: integrality of the limiting interface in the strong coupling regime

We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features two order parameters: a reduced $\Q$-tensor field $\Q$, describing the nematic liquid crystal component, and a vector field $\M$, accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between $\Q$ and $\M$. It has been shown in~\cite{CDS1,CDS2} that, as a small parameter $\eps$ tends to zero, the energy of critical pairs $(\Q_\eps,\,\M_\eps)$ concentrates on \emph{distinct} singular sets: a finite set of points for the $\Q_\eps$-component and the support of a $\H^1$-rectifiable varifold for the $\M_\eps$-component, with first variation supported on the singular set for the $\Q_\eps$-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a material constant, such a limiting varifold has integer multiplicity.

math.AP↗

arXiv2026

Singular extension of critical Sobolev mappings with values into complete Riemannian manifolds

Relying on a recent criterion, due to A.~Petrunin [13], to check if a complete, non-compact, Riemannian manifold admits an isometric embedding with positive reach into a Euclidean space, we extend to manifolds with such a property the singular extension results of B.~Bulanyi and J.~Van~Schaftingen [5] for maps in the critical, nonlinear Sobolev space $W^{m/(m+1),m+1}\left(X^m,\mathcal{N}\right)$, where $m \in \mathbb{N} \setminus \{0\}$, $\mathcal{N}$ is a compact Riemannian manifold, and $X^m$ is either the sphere $\mathbb{S}^m = \partial \mathbb{B}^{m+1}_+$, the plane $\mathbb{R}^m$, or again $\mathbb{S}^m$ but seen as the boundary sphere of the Poincaré ball model of the hyperbolic space $\mathbb{H}^{m+1}$. As in [5], we obtain that the extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate. Finally, we provide some illustrative examples of non-compact target manifolds to which our results apply.

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arXiv2026

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

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}.

math.AP↗

arXiv2026

The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics. Part~I: Energy estimates and compactness results

We study 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, the largest part of which is carried by the $\Q$-component, includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. In this article and in the companion paper~\cite{CDS2}, we analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. In this paper, we prove that the (rescaled) energy density for the $\Q$-component, concentrates, to leading order, on a finite number of singular points. Moreover, we prove energy estimates and compactness results that will be crucially used in~\cite{CDS2} to determine the structure of the energy concentration set for the $\M$-component as well as the relationship between the two singular sets.

math.AP↗

arXiv2026

Liftings of Sobolev maps into closed Riemannian manifolds via double coverings and minimal connections relative to planar sets, with an application to ferronematics

We consider Sobolev maps from a planar domain into a closed Riemannian manifold and their BV liftings via a double covering of the target. We establish a sharp lower bound on the jump length of the lifting, expressed in terms of a geometric quantity: the minimal connection, relative to the domain, of the non-orientable singularities. As an application, we analyse minimisers of a two-dimensional model of ferronematics under ``mixed'' boundary conditions -- that is, Dirichlet conditions for the liquid crystal order parameter and Neumann conditions for the magnetisation vector.

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arXiv2023

Manifold-constrained free discontinuity problems and Sobolev approximation

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $Ω\subset \R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.

math.AP↗

arXiv2023

The Yang-Mills-Higgs functional on complex line bundles: $Γ$-convergence and the London equation

We consider the Abelian Yang-Mills-Higgs functional, in the non-self dual scaling, on a complex line bundle over a closed Riemannian manifold of dimension $n\geq 3$. This functional is the natural generalisation of the Ginzburg-Landau model for superconductivity to the non-Euclidean setting. We prove a $Γ$-convergence result, in the strongly repulsive limit, on the functional rescaled by the logarithm of the coupling parameter. As a corollary, we prove that the energy of minimisers concentrates on an area-minimising surface of dimension $n-2$, while the curvature of minimisers converges to a solution of the London equation.

math.AP↗

arXiv2023

The Yang-Mills-Higgs functional on complex line bundles: asymptotics for critical points

We consider a gauge-invariant Ginzburg-Landau functional (also known as Abelian Yang-Mills-Higgs model) on Hermitian line bundles over closed Riemannian manifolds of dimension $n \geq 3$. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the non-self dual scaling, as the coupling parameter tends to zero. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, %independently of the gauge andthanks to a suitable monotonicity formula,we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, concentrate towards the weight measure of a stationary, rectifiable varifold of codimension~2.

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