Radu Ignat
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Previous Organizations / Institutes
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Radu Ignat has a total of 29 co-authors. Here are the top co-authors by number of shared publications:
- Luc Nguyen
Mathematical Institute and St Edmund Hall, University of Oxford, Oxford, UK
12 shared publications - Arghir Zarnescu
Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Bizkaia, Spain
8 shared publications - Valeriy Slastikov
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
7 shared publications - Roger Moser
University of Bath BA2 7AY Bath UK
5 shared publications - Felix Otto
Institute of Solid State Physics Friedrich Schiller University Jena Jena Germany
3 shared publications - Mickaël Nahon
CNRS, Grenoble INP, LJK, Univ. Grenoble Alpes, Grenoble, France
3 shared publications
Latest Publications
- Open Access
Asymptotic minimality of 1D 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 R 2 -valued 1-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially onedimensional.
Open AccessAsymptotic minimality of 1D 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 R 2 -valued 1-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially onedimensional.
- Open Access
- 21 July 2025
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 363 (G9) , 887-891
A short proof of the regularity for the eikonal equationWe give a short and self-contained proof of the interior regularity of solutions to the eikonal equation in an open set in dimension under the assumption that is pointwise differentiable in .- 21 July 2025
- Published by MathDoc/Centre Mersenne in Comptes Rendus Mathematique
- Vol. 363 (G9) , 887-891
Open AccessA short proof of the regularity for the eikonal equationWe give a short and self-contained proof of the interior regularity of solutions to the eikonal equation in an open set in dimension under the assumption that is pointwise differentiable in . - Open Access
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_σ(\boldsymbol{m}) = \int_{\mathbb R^2} \bigg(\frac 12 |\nabla\boldsymbol{m}|^2 +σ^2 \boldsymbol{ m} \cdot \nabla \times\boldsymbol{m} +σ^2 m_3^2 \bigg)\, dx\,, \end{align*} for $0<σ\ll 1$, with prescribed topological degree \begin{align*} Q(\boldsymbol{m})=\frac{1}{4π} \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 Möbius maps: we establish in particular that they are localized at scale of order $1/|\ln(σ^2)|$. 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.Open AccessThe 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_σ(\boldsymbol{m}) = \int_{\mathbb R^2} \bigg(\frac 12 |\nabla\boldsymbol{m}|^2 +σ^2 \boldsymbol{ m} \cdot \nabla \times\boldsymbol{m} +σ^2 m_3^2 \bigg)\, dx\,, \end{align*} for $0<σ\ll 1$, with prescribed topological degree \begin{align*} Q(\boldsymbol{m})=\frac{1}{4π} \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 Möbius maps: we establish in particular that they are localized at scale of order $1/|\ln(σ^2)|$. 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. - Open Access
- 24 January 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (1) , 1-35
Minimality of Vortex Solutions to Ginzburg–Landau Type Systems for Gradient Fields in the Unit Ball in Dimension $N\ge 4$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 $$N \ge 4$$ and with respect to its boundary value. A similar result is also prove in a model for $$\mathbb {S}^N$$ -valued maps arising 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 $$N \ge 7$$ . The second method uses a symmetrization procedure for gradient fields such that the $$L^2$$ -norm is invariant while the $$L^p$$ -norm with $$2< p < \infty $$ and the $$H^1$$ -norm are lowered.- 24 January 2025
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 249 (1) , 1-35
Open AccessMinimality of Vortex Solutions to Ginzburg–Landau Type Systems for Gradient Fields in the Unit Ball in Dimension $N\ge 4$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 $$N \ge 4$$ and with respect to its boundary value. A similar result is also prove in a model for $$\mathbb {S}^N$$ -valued maps arising 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 $$N \ge 7$$ . The second method uses a symmetrization procedure for gradient fields such that the $$L^2$$ -norm is invariant while the $$L^p$$ -norm with $$2< p < \infty $$ and the $$H^1$$ -norm are lowered. - Open AccessMinimality 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 ≥ 4 and with respect to its boundary value. A similar result is also proved in a model for S N -valued maps arising 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 ≥ 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 < ∞, and the H 1 -norm are lowered.
Open AccessMinimality 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 ≥ 4 and with respect to its boundary value. A similar result is also proved in a model for S N -valued maps arising 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 ≥ 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 < ∞, and the H 1 -norm are lowered.
- Open AccessA SHORT PROOF OF THE $C^{1,1}$ REGULARITY FOR THE EIKONAL EQUATIONWe give a short and self-contained proof of the interior $C^{1,1}$ regularity of solutions φ:Ω→ℝ to the eikonal equation |∇φ|=1 in an open set Ω⊂R^N in dimension N≥1 under the assumption that φ is pointwise differentiable in Ω.Open AccessA SHORT PROOF OF THE $C^{1,1}$ REGULARITY FOR THE EIKONAL EQUATIONWe give a short and self-contained proof of the interior $C^{1,1}$ regularity of solutions φ:Ω→ℝ to the eikonal equation |∇φ|=1 in an open set Ω⊂R^N in dimension N≥1 under the assumption that φ is pointwise differentiable in Ω.
- Open AccessA short proof of the $\mathcal C^{1,1}$ regularity for the eikonal equationWe give a short and self-contained proof of the interior $\mathcal C^{1,1}$ regularity of solutions $\varphi:Ω\to \mathbb{R}$ to the eikonal equation $|\nabla \varphi|=1$ in an open set $Ω\subset \mathbb{R}^{N}$ in dimension $N\geq 1$ under the assumption that $\varphi$ is pointwise differentiable in $Ω$.Open AccessA short proof of the $\mathcal C^{1,1}$ regularity for the eikonal equationWe give a short and self-contained proof of the interior $\mathcal C^{1,1}$ regularity of solutions $\varphi:Ω\to \mathbb{R}$ to the eikonal equation $|\nabla \varphi|=1$ in an open set $Ω\subset \mathbb{R}^{N}$ in dimension $N\geq 1$ under the assumption that $\varphi$ is pointwise differentiable in $Ω$.
- Open Access
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 $Γ$-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.
Open AccessRenormalised 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 $Γ$-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.
- 17 January 2024
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 63 (2) , 1-20
Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality
- 17 January 2024
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 63 (2) , 1-20
Vortex sheet solutions for the Ginzburg–Landau system in cylinders: symmetry and global minimality
- Open AccessMinimality 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.Open AccessMinimality 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.