arXiv subjects

Xavier Lamy

Publications and source records attributed to Xavier Lamy.

At least 19 recordsLinked to original sources

arXiv2026

Some lifting and approximation properties for maps in $W^{1,2}(\mathbb{B}^3;\mathbb{S}^2)$

Smooth maps $u\colon\mathbb B^3\to\mathbb S^2$ can be lifted to $\hat u\colon\mathbb B^3\to\mathbb S^3$ using the Hopf fibration $h\colon \mathbb S^3\to\mathbb S^2$ via the factorization $u=h\circ\hat u$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^*\omega_{\mathbb S^2}$, and deduce a smooth approximation property preserving the constraint $u^*\omega_{\mathbb S^2}=d\eta$.

math.AP↗

arXiv2026

On Aviles-Giga limit states with $L^p$ entropy productions

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colon\Omega \subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(\Omega),\quad |m|=1\text{ a.e. in }\Omega\,, \end{align*} whose entropy productions $\mathrm{div}\,\Phi(m)$ are Radon measures in $\Omega$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $\Phi\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,\Phi(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.

math.AP↗

arXiv2026

Hyperbolic regularization effects for degenerate elliptic equations

This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on the differential inclusions point of view, we establish a pointwise gradient localization theorem and we show that the singular set of nondifferentiability points of $u$ is $\mathcal{H}^1$-negligible. As a consequence, we derive new sharp partial $C^1$ regularity results under the assumption that $G$ is degenerate only on curves. This is done by exploiting the hyperbolic structure of the equation along these curves, where the loss of regularity is compensated using tools from the theories of Hamilton-Jacobi equations and scalar conservation laws. Our analysis recovers and extends all the previously known results, where the degeneracy set was required to be zero-dimensional.

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arXiv2025

Rectifiability of entropy productions for weak solutions of the 2D eikonal equation with supercritical regularity

Weak solutions $m\colon\Omega\subset\mathbb{R}^2\to\mathbb{R}^2$ of the eikonal equation \begin{align*} |m|=1\text{ a.e. and }\mathrm{div}\: m =0\,, \end{align*} arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles-Giga energy. The distributions $\mu_\Phi=\mathrm{div}\,\Phi(m)$, defined for a class of smooth vector fields $\Phi$ called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of $m$, as they do if $m$ has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption \begin{align*} m\in B^{1/p}_{p,\infty} \quad \Leftrightarrow \quad \sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\|m(\cdot +h)-m\|_{L^p }}{|h|^{1/p}} <\infty\,, \end{align*} for any $1\leq p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open.

math.AP↗

arXiv2025

Compensation effects for anisotropic energies of two-dimensional unit vector fields

We study the highly anisotropic energy of two-dimensional unit vector fields given by \begin{align*} E_\epsilon(u)= \int_{\Omega} (\mathrm{div}\,u)^2 + \epsilon(\mathrm{curl}\,u)^2\, dx\,, \quad u\colon\Omega\subset\mathbb R^2\to\mathbb S^1\, \end{align*} in the limit $\epsilon\to 0$. This energy clearly loses control on the full gradient of $u$ as $\epsilon\to 0$, but, adapting tools from hyperbolic conservations laws, we show that it still controls derivatives of order 1/2. In particular, any bounded energy sequence $E_\epsilon(u_\epsilon)\leq C$ is compact in $W^{s,3}_{\mathrm{loc}}(\Omega)$ for $s<1/2$. Moreover, this order 1/2 of differentiability is optimal, in the sense that any map $u\in W^{1/2,4}(\Omega;\mathbb S^1)$ is a limit of a bounded energy sequence. We also establish compactness of boundary traces in $L^1(\partial\Omega)$, and characterize the $\Gamma$-limit in the simpler case of maps of a single variable and in the case of a thin-film model.

math.AP↗

arXiv2025

Global minimality of the Hopf map in the Faddeev-Skyrme model with large coupling constant

We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.

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↗

arXiv2025

Another regularizing property of the 2D eikonal equation

A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colon\Omega\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(\Omega)$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1/3,3}$, then $m$ is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption $m\in B^{1/3}_{p,\infty}$ for $p>3$. In this note we establish that regularizing effect in the borderline case $p=6$, above which the Besov regularity assumption implies continuity. If the domain is a disk and $m$ satisfies tangent boundary conditions, we also prove this for $p$ slightly below $6$.

math.AP↗

arXiv2025

Interaction energies in nematic liquid crystal suspensions

We establish, as $\rho\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $\rho$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_\rho = \sum_j \mu_j -4\pi\rho \sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(\rho)\,, \end{align*} where $\mu_j$ is the minimal energy after zooming in at scale $\rho$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.

math.AP↗

arXiv2024

On the existence of degenerate solutions of the two-dimensional $H$-system

We consider entire solutions $\omega\in\dot H^1(\mathbb R^2;\mathbb R^3)$ of the $H$-system $\Delta\omega=2\omega_x\wedge\omega_y,$ which we refer to as bubbles. Surprisingly, and contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a complete algebraic characterization of degenerate bubbles.

math.AP↗

arXiv2024

On $C^1$ regularity for degenerate elliptic equations in the plane

We show that Lipschitz solutions $u$ of $\mathrm{div}\, G(\nabla u)=0$ in $B_1\subset\mathbb R^2$ are $C^1$, for strictly monotone vector fields $G\in C^0(\mathbb R^2;\mathbb R^2)$ satisfying a mild ellipticity condition. If $G=\nabla F$ for a strictly convex function $F$, and $0\leq \lambda(\xi)\leq \Lambda(\xi)$ are the two eigenvalues of $\nabla^2 F(\xi)$, our assumption is that the set $\lbrace\lambda=0\rbrace \cap \lbrace \Lambda=\infty\rbrace$, where ellipticity degenerates $both$ from below and from above, is finite. This extends results by De Silva and Savin (Duke Math. J. 151, No. 3, p.487-532, 2010), which assumed either that set empty, or the larger set $\lbrace \lambda=0\rbrace$ finite. Our main new input is to transfer estimates in $\lbrace \lambda > 0 \rbrace $ to estimates in $\lbrace \Lambda <\infty\rbrace$ by means of a conjugate equation. When $G$ is not a gradient, the ellipticity assumption needs to be interpreted in a specific way, and we highlight the nontrivial effect of the antisymmetric part of $\nabla G$.

math.AP↗

arXiv2024

On regularity and rigidity of $2\times 2$ differential inclusions into non-elliptic curves

We study differential inclusions $Du\in \Pi$ in an open set $\Omega\subset\mathbb R^2$, where $\Pi\subset \mathbb R^{2\times 2}$ is a compact connected $C^2$ curve without rank-one connections, but non-elliptic: tangent lines to $\Pi$ may have rank-one connections, so that classical regularity and rigidity results do not apply. For a wide class of such curves $\Pi$, we show that $Du$ is locally Lipschitz outside a discrete set, and is rigidly characterized around each singularity. Moreover, in the partially elliptic case where at least one tangent line to $\Pi$ has no rank-one connections, or under some topological restrictions on the tangent bundle of $\Pi$, there are no singularities. This goes well beyond previously known particular cases related to Burgers' equation and to the Aviles-Giga functional. The key is the identification and appropriate use of a general underlying structure: an infinite family of conservation laws, called entropy productions in reference to the theory of scalar conservation laws.

math.AP↗

arXiv2024

Optimal quantitative stability of the M\"obius group of the sphere in all dimensions

In any dimension $n\geq 3$, we prove an optimal stability estimate for the M\"obius group among maps $u\colon \mathbb S^{n-1} \to \mathbb R^n$, of the form $\inf_{\lambda>0,\phi\in \mathrm{M\"ob}(\mathbb S^{n-1})} \int_{\mathbb S^{n-1}}\left|\frac 1\lambda \nabla_{T} u -\nabla_{ T}\phi\right|^{n-1} d\mathcal H^{n-1} \leq C_n \mathcal E_{n-1}(u).$ Here, $\mathcal E_{n-1}(u)$ is a conformally invariant deficit which measures simultaneously lack of conformality and the deviation of $u(\mathbb S^{n-1})$ from being a round sphere in an isoperimetric sense. This entails in particular the following qualitative statement: sequences with vanishing deficit, once appropriately normalized by the action of the M\"obius group, are compact. Both the qualitative and the quantitative results are new for all dimensions $n\geq 4$.

math.DG↗

arXiv2023

A symmetry breaking phenomenon for anisotropic harmonic maps from a 2D annulus into $\mathbb S^1$

In a two dimensional annulus $A_\rho=\{x\in \mathbb R^2: \rho<|x|<1\}$, $\rho\in (0,1)$, we characterize $0$-homogeneous minimizers, in $H^1(A_\rho;\mathbb S^1)$ with respect to their own boundary conditions, of the anisotropic energy \begin{equation*} E_\delta(u)=\int_{A_\rho} |\nabla u|^2 +\delta \left( (\nabla\cdot u)^2-(\nabla\times u)^2\right) \, dx,\quad \delta\in (-1,1). \end{equation*} Even for a small anisotropy $0<|\delta|\ll 1$, we exhibit qualitative properties very different from the isotropic case $\delta=0$. In particular, $0$-homogeneous critical points of degree $d\notin \lbrace 0,1,2\rbrace$ are always local minimizers, but in thick annuli ($\rho\ll 1$) they are not minimizers: the $0$-homogeneous symmetry is broken. One corollary is that entire solutions to the anisotropic Ginzburg-Landau system have a far-field behavior very different from the isotropic case studied by Brezis, Merle and Rivi\`ere. The tools we use include: ODE and variational arguments; asymptotic expansions, interpolation inequalities and explicit computations involving near-optimizers of these inequalities for proving that $0$-homogeneous critical points are not minimizers in thick annuli.

math.AP↗

arXiv2023

Sharp quantitative stability of the M\"obius group among sphere-valued maps in arbitrary dimension

In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal. 239(1):219-299, 2021), with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to M\"obius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

math.AP↗

arXiv2022

Generation of vortices in the Ginzburg-Landau heat flow

We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.

math.AP↗

arXiv2022

On Lebesgue points of entropy solutions to the eikonal equation

We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless they share with BV functions, several of their fine properties: we show in particular that the set of non-Lebesgue points has co-dimension at least one.

math.AP↗

arXiv2022

Stability of the vortex in micromagnetics and related models

We consider line-energy models of Ginzburg-Landau type in a two-dimensional simply-connected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of $C^{1,1}$ domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.

math.AP↗