arXiv subjects

Radu Ignat

Publications and source records attributed to Radu Ignat.

At least 19 recordsLinked to original sources

arXiv2026

The Ginzburg-Landau system with general potential: maximum principle and gradient estimates

We study critical points of the Ginzburg-Landau energy functional $$\mathcal{F}_\varepsilon[u] = \int_\Omega \Big[ \frac{1}{2}|\nabla u|^2 + \frac{1}{2\varepsilon^2} W(1 - |u|^2)\Big]\,dx, \quad u \in H^1(\Omega, \mathbb{R}^N),$$ with $\Omega \subset \mathbb{R}^M$, $\varepsilon>0$, $M,N \geq 2$ and general conditions on the non-negative potential $W$ allowing for super-quadratic behaviour near its zero set. Under a Dirichlet boundary data of unit-length on $\partial \Omega$, we prove the following maximum principle: every critical point $u_\varepsilon$ satisfies the global uniform bound $|u_\varepsilon| \leq 1$ in $\Omega$. Furthermore, if a family of critical points $(u_\varepsilon)$ converges (in energy) to a smooth $\mathbb{S}^{N-1}$-valued harmonic map in the limit $\varepsilon \to 0$, then we prove global uniform bounds for $(\Delta u_\varepsilon)_{\varepsilon>0}$ in $\Omega$ and, in particular, global H\"older convergence of the gradients $(\nabla u_\varepsilon)$ in $\Omega$ as $\varepsilon \to 0$.

math.AP↗

arXiv2025

Asymptotic minimality of one-dimensional transition profiles in Aviles-Giga type models: an approach via 1-currents

For vector fields on a two-dimensional domain, we study the asymptotic behaviour of Modica-Mortola (or Allen-Cahn) type functionals under the assumption that the divergence converges to $0$ at a certain rate, which effectively produces a model of Aviles-Giga type. This problem will typically give rise to transition layers, which degenerate into discontinuities in the limit. We analyse the energy concentration at these discontinuities and the corresponding transition profiles. We derive an estimate for the energy concentration in terms of a novel geometric variational problem involving the notion of $\mathbb{R}^2$-valued $1$-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially one-dimensional.

math.AP↗

arXiv2025

The conformal limit for bimerons in easy-plane chiral magnets

We study minimizers $\boldsymbol{m}\colon \mathbb R^2\to\mathbb S^2$ of the energy functional \begin{align*} E_\sigma(\boldsymbol{m}) = \int_{\mathbb R^2} \bigg(\frac 12 |\nabla\boldsymbol{m}|^2 +\sigma^2 \boldsymbol{ m} \cdot \nabla \times\boldsymbol{m} +\sigma^2 m_3^2 \bigg)\, dx\,, \end{align*} for $0<\sigma\ll 1$, with prescribed topological degree \begin{align*} Q(\boldsymbol{m})=\frac{1}{4\pi} \int_{\mathbb R^2}\boldsymbol{m} \cdot \partial_1 \boldsymbol{m}\times\partial_2\boldsymbol{m}\, dx =\pm 1\,. \end{align*} This model arises in thin ferromagnetic films with Dzyaloshinskii-Moriya interaction and easy-plane anisotropy, where these minimizers represent bimeron configurations. We prove their existence, and describe them precisely as perturbations of specific M\"obius maps: we establish in particular that they are localized at scale of order $1/|\ln(\sigma^2)|$. The proof follows a strategy introduced by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal., 2021) for a similar model with easy-axis anisotropy, but requires several adaptations to deal with the less coercive easy-plane anisotropy and different symmetry properties.

math.AP↗

arXiv2024

A short proof of the $\mathcal C^{1,1}$ regularity for the eikonal equation

We give a short and self-contained proof of the interior $\mathcal C^{1,1}$ regularity of solutions $\varphi:\Omega \to \mathbb{R}$ to the eikonal equation $|\nabla \varphi|=1$ in an open set $\Omega\subset \mathbb{R}^{N}$ in dimension $N\geq 1$ under the assumption that $\varphi$ is pointwise differentiable in $\Omega$.

math.AP↗

arXiv2024

Renormalised energy between boundary vortices in thin-film micromagnetics with Dzyaloshinskii-Moriya interaction

We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg-Landau type model in terms of the averaged magnetization in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a $\Gamma$-convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.

math.AP↗

arXiv2023

Minimality of vortex solutions to Ginzburg--Landau type systems for gradient fields in the unit ball in dimension $N\geq 4$

We prove that the degree-one vortex solution is the unique minimizer for the Ginzburg--Landau functional for gradient fields (that is, the Aviles--Giga model) in the unit ball $B^N$ in dimension $N \geq 4$ and with respect to its boundary value. A similar result is also proved for $\mathbb{S}^N$-valued maps in the theory of micromagnetics. Two methods are presented. The first method is an extension of the analogous technique previously used to treat the unconstrained Ginzburg--Landau functional in dimension $N \geq 7$. The second method uses a symmetrization procedure for gradient fields such that the $L^2$-norm is invariant while the $L^p$-norm, $2 < p < \infty$, and the $H^1$-norm are lowered.

math.AP↗

arXiv2023

Vortex sheet solutions for the Ginzburg-Landau system in cylinders: symmetry and global minimality

We consider the Ginzburg-Landau energy $E_\epsilon$ for $\mathbb{R}^M$-valued maps defined in a cylinder shape domain $B^N\times (0,1)^n$ satisfying a degree-one vortex boundary condition on $\partial B^N\times (0,1)^n$ in dimensions $M\geq N\geq 2$ and $n\geq 1$. The aim is to study the radial symmetry of global minimizers of this variational problem. We prove the following: if $N\geq 7$, then for every $\epsilon>0$, there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution $u_\epsilon(x,z)=(f_\epsilon(|x|) \frac{x}{|x|}, 0_{\mathbb{R}^{M-N}})$, $\forall x\in B^N$ that is invariant in $z\in (0,1)^n$. If $2\leq N \leq 6$ and $M\geq N+1$, the following dichotomy occurs between escaping and non-escaping solutions: there exists $\epsilon_N>0$ such that $\bullet$ if $\epsilon\in (0, \epsilon_N)$, then every global minimizer is an escaping radially symmetric vortex sheet solution of the form $R \tilde u_\epsilon$ where $\tilde{u}_{\epsilon}(x,z)=(\tilde{f}_{\epsilon}(|x|) \frac{x}{|x|}, 0_{\mathbb{R}^{M-N-1}}, g_{\epsilon}(|x|))$ is invariant in $z$-direction with $g_\epsilon>0$ in $(0,1)$ and $R\in O(M)$ is an orthogonal transformation keeping invariant the space $\mathbb{R}^N\times \{0_{\mathbb{R}^{M-N}}\}$; $\bullet$ if $\epsilon\geq \epsilon_N$, then the non-escaping radially symmetric vortex sheet solution $u_\epsilon(x,z)=(f_\epsilon(|x|) \frac{x}{|x|}, 0_{\mathbb{R}^{M-N}})$, $\forall x\in B^N, z\in (0,1)^n$ is the unique global minimizer; moreover, there are no bounded escaping solutions in this case. We also discuss the problem of vortex sheet $\mathbb{S}^{M-1}$-valued harmonic maps.

math.AP↗

arXiv2022

An effective model for boundary vortices in thin-film micromagnetics

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of "global Jacobian" detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $\Gamma$-convergence expansion at the second order. The second order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

math.AP↗

arXiv2022

Asymptotic stability of precessing domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction

We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via $\Gamma$-convergence that determines a limit functional $E$ defined for maps $m:\mathbb{R}\to \mathbb{S}^2$ in the direction $e_1$ of the nanowire. The energy functional $E$ is invariant under translations in $e_1$ and rotations about the axis $e_1$. We fully classify the critical points of finite energy $E$ when a transition between $-e_1$ and $e_1$ is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau-Lifshitz-Gilbert equation associated to $E$ under the effect of an applied magnetic field $h(t)e_1$ depending on the time variable $t$ gives rise to the so-called precessing domain wall. Our main result proves the asymptotic stability of precessing domain walls for small $h$ in $L^\infty([0, +\infty))$ and small $H^1(\mathbb{R})$ perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional $E$.

math.AP↗

arXiv2021

Uniqueness result for a weighted pendulum equation modeling domain walls in notched ferromagnetic nanowires

We prove an existence and uniqueness result for solutions $\varphi$ to a weighted pendulum equation in $\mathbb{R}$ where the weight is non-smooth and coercive. We also establish (in)stability results for $\varphi$ according to the monotonicity of the weight. These results are applied in a reduced model for thin ferromagnetic nanowires with notches to obtain existence, uniqueness and stability of domain walls connecting two opposite directions of the magnetization.

math.CA↗

arXiv2021

Local minimality of $\mathbb{R}^N$-valued and $\mathbb{S}^N$-valued Ginzburg-Landau vortex solutions in the unit ball $B^N$

We study the existence, uniqueness and minimality of critical points of the form $m_{\varepsilon,\eta}(x) = (f_{\varepsilon,\eta}(|x|)\frac{x}{|x|}, g_{\varepsilon,\eta}(|x|))$ of the functional \[ E_{\varepsilon,\eta}[m] = \int_{B^N} \Big[\frac{1}{2} |\nabla m|^2 + \frac{1}{2\varepsilon^2} (1 - |m|^2)^2 + \frac{1}{2\eta^2} m_{N+1}^2\Big]\,dx \] for $m=(m_1, \dots, m_N, m_{N+1}) \in H^1(B^N,\mathbb{R}^{N+1})$ with $m(x) = (x,0)$ on $\partial B^N$. We establish a necessary and sufficient condition on the dimension $N$ and the parameters $\varepsilon$ and $\eta$ for the existence of an escaping vortex solution $(f_{\varepsilon,\eta}, g_{\varepsilon,\eta})$ with $g_{\varepsilon,\eta}> 0$. We also establish its uniqueness and local minimality. In the limiting case $\eta = 0$, we prove the local minimality of the degree-one vortex solution for the Ginzburg-Landau (GL) energy for every $\varepsilon > 0$ and $N \geq 2$. Similarly, when $\varepsilon = 0$, we prove the local minimality of the degree-one escaping vortex solution to an $\mathbb{S}^N$-valued GL model arising in micromagnetics for every $\eta > 0$ and $2 \leq N \leq 6$.

math.AP↗

arXiv2020

Variational methods for a singular SPDE yielding the universality of the magnetization ripple

The magnetization ripple is a microstructure formed in thin ferromagnetic films. It can be described by minimizers of a nonconvex energy functional leading to a nonlocal and nonlinear elliptic SPDE in two dimensions driven by white noise, which is singular. We address the universal character of the magnetization ripple using variational methods based on $\Gamma$-convergence. Due to the infinite energy of the system, the (random) energy functional has to be renormalized. Using the topology of $\Gamma$-convergence, we give a sense to the law of the renormalized functional that is independent of the way white noise is approximated. More precisely, this universality holds in the class of (not necessarily Gaussian) approximations to white noise satisfying the spectral gap inequality, which allows us to obtain sharp stochastic estimates. As a corollary, we obtain the existence of minimizers with optimal regularity.

math.PR↗

arXiv2020

Separation of domain walls with nonlocal interaction and their renormalised energy by $\Gamma$-convergence in thin ferromagnetic films

We analyse two variants of a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter $\epsilon > 0$. The model gives rise to transition layers, called N\'eel walls, and we study their behaviour in the limit $\epsilon \to 0$. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between N\'eel walls to leading order. But while Ginzburg-Landau vortices show attraction for degrees of the same sign and repulsion for degrees of opposite signs, the pattern is reversed in this model. In a previous paper, we determined the renormalised energy for one of the models studied here under the assumption that the N\'eel walls stay separated from each other. In this paper, we present a deeper analysis that in particular removes this assumption. The theory gives rise to an effective variational problem for the positions of the walls, encapsulated in a $\Gamma$-convergence result. In the second part of the paper, we turn our attention to another, more physical model, including an anisotropy term. We show that it permits a similar theory, but the anisotropy changes the renormalised energy in unexpected ways and requires different methods to find it.

math.AP↗

arXiv2019

Global Jacobian and $\Gamma$-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of $2D$ Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a $2D$ map $u$. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a $2D$ Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of $u$ at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by $\Gamma$-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.

math.AP↗

arXiv2019

Renormalized energy between vortices in some Ginzburg-Landau models on 2-dimensional Riemannian manifolds

We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $\mathfrak{g}$ of $S$ is different than $1$. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the $2\mathfrak{g}$-dimensional space of harmonic $1$-forms on $S$ if $\mathfrak{g}\geq 1$. Our second main result determines the interaction energy (called renormalized energy) between vortex points as a $\Gamma$-limit (at the second order) as $\varepsilon\to 0$. The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of $S$ as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg-Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces $S\subset \mathbb{R}^3$, in particular, to a physical model for non-tangent maps to $S$ coming from micromagnetics.

math.AP↗

arXiv2019

Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals

We consider a variational two-dimensional Landau-de Gennes model in the theory of nematic liquid crystals in a disk of radius $R$. We prove that under a symmetric boundary condition carrying a topological defect of degree $\frac{k}{2}$ for some given {\bf even} non-zero integer $k$, there are exactly two minimizers for all large enough $R$. We show that the minimizers do not inherit the full symmetry structure of the energy functional and the boundary data. We further show that there are at least five symmetric critical points.

math.AP↗

arXiv2019

A necessary condition in a De Giorgi type conjecture for elliptic systems in infinite strips

Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at infinity, we show that every map $u:\mathbb{R}\times\omega\to\mathbb{R}^N$ with \[ \int_{\mathbb{R}\times\omega}\big(\lvert\nabla u\rvert^2+W(u)\big)\mathop{}\mathopen{}\mathrm{d} x_1\mathop{}\mathopen{}\mathrm{d}x'<+\infty\] has a limit $u^\pm\in\{W=0\}$ as $x_1\to\pm\infty$. The convergence holds in $L^2(\omega)$ and almost everywhere in $\omega$. We also prove a similar result for more general potentials $W$ in the case where the considered maps $u$ are divergence-free in $\mathbb{R}\times\omega$ with $\omega$ being the $(d-1)$-torus and $N=d$.

math.AP↗

arXiv2019

Global uniform estimate for the modulus of $2D$ Ginzburg-Landau vortexless solutions with asymptotically infinite boundary energy

For $\varepsilon>0$, let $u_\varepsilon:\Omega\to \mathbb R^2$ be a solution of the Ginzburg-Landau system $$-\Delta u_\varepsilon=\frac 1{\varepsilon^2} u_\varepsilon (1-|u_\varepsilon|^2)$$ in a Lipschitz bounded domain $\Omega$. In an energy regime that excludes interior vortices, we prove that $1-|u_\varepsilon|$ is uniformly estimated by a positive power of $\varepsilon$ $globally$ in $\Omega$ provided that the energy of $u_\varepsilon$ at the boundary $\partial \Omega$ does not grow faster than $\varepsilon^{-\alpha}$ with $\alpha\in (0,1)$.

math.AP↗