arXiv subjects

Antonin Monteil

Publications and source records attributed to Antonin Monteil.

12 recordsLinked to original sources

arXiv2024

Magnetic skyrmions under confinement

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy, with a possible addition of perpendicular magnetic anisotropy. Under Dirichlet boundary conditions that are motivated by the asymptotic treatment of the stray field energy in the thin film limit we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the $Γ$-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications. We provide explicit calculations of the renormalized energy for a number of basic domain geometries.

math.AP↗

arXiv2024

Mass concentration in rescaled first order integral functionals

We consider first order local minimization problems of the form $\min \int_{\mathbb{R}^N}f(u,\nabla u)$ under a mass constraint $\int_{\mathbb{R}^N}u=m$. We prove that the minimal energy function $H(m)$ is always concave, and that relevant rescalings of the energy, depending on a small parameter $\varepsilon$, $Γ$-converge towards the $H$-mass, defined for atomic measures $\sum_i m_iδ_{x_i}$ as $\sum_i H(m_i)$. We also consider Lagrangians depending on $\varepsilon$, as well as space-inhomogeneous Lagrangians and $H$-masses. Our result holds under mild assumptions on $f$, and covers in particular $α$-masses in any dimension $N\geq 2$ for exponents $α$ above a critical threshold, and all concave $H$-masses in dimension $N=1$. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

math.AP↗

arXiv2021

Symmetry properties of minimizers of a perturbed Dirichlet energy with a boundary penalization

We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minimizers with different penalizations. Finally, we apply these results to a problem on a ball and show radial symmetry and monotonicity of minimizers. In dimension $N=2$ our results can be applied to the Oseen--Frank energy for nematic liquid crystals and micromagnetic energy in a thin-film regime.

math.AP↗

arXiv2021

Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold

We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and Hélein for the circle $\mathbb{S}^1$. We also obtain $Γ$-convergence results and uniform Marcinkiewicz weak $L^2$ or Lorentz $L^2$ estimates on the derivatives. We prove that solutions to the corresponding Euler-Lagrange equation converge uniformly to the constraint and converge to harmonic maps away from singularities.

math.AP↗

arXiv2021

Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and Hélein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress-energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.

math.AP↗

arXiv2019

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

Given a bounded Lipschitz domain $ω\subset\mathbb{R}^{d-1}$ 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 $u:\mathbb{R}\timesω\to\mathbb{R}^N$ with \[ \int_{\mathbb{R}\timesω}\big(\lvert\nabla u\rvert^2+W(u)\big)\mathop{}\mathopen{}\mathrm{d} x_1\mathop{}\mathopen{}\mathrm{d}x'<+\infty\] has a limit $u^\pm\in\{W=0\}$ as $x_1\to\pm\infty$. The convergence holds in $L^2(ω)$ and almost everywhere in $ω$. We also prove a similar result for more general potentials $W$ in the case where the considered maps $u$ are divergence-free in $\mathbb{R}\timesω$ with $ω$ being the $(d-1)$-torus and $N=d$.

math.AP↗

arXiv2018

A DeGiorgi type conjecture for minimal solutions to a nonlinear Stokes equation

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation $$ \begin{cases} -Δu+\nabla W(u)=\nabla p&\text{in }\mathbb{R}^d,\\ \nabla\cdot u=0&\text{in }\mathbb{R}^d, \end{cases} $$ which are periodic in the $d-1$ last variables (living on the torus $\mathbb{T}^{d-1}$) and globally minimize the corresponding energy in $Ω=\mathbb{R}\times \mathbb{T}^{d-1}$, i.e., $$ E(u)=\int_Ω \frac12 |\nabla u|^2+W(u)\, dx, \quad \nabla\cdot u=0. $$ Namely, we determine a class of nonlinear potentials $W\geq 0$ such that any global minimizer $u$ of $E$ connecting two zeros of $W$ as $x_1\to\pm\infty$ is one-dimensional, i.e., $u$ depends only on the $x_1$ variable. In particular, this class includes in dimension $d=2$ the nonlinearities $W=w^2$ with $w$ being an harmonic function or a solution to the wave equation, while in dimension $d\geq 3$, this class contains a perturbation of the Ginzburg-Landau potential as well as potentials $W$ having $d+1$ wells with prescribed transition cost between the wells. For that, we develop a theory of calibrations relying on the notion of entropy (coming from scalar conservation laws). We also study the problem of the existence of global minimizers of $E$ for general potentials $W$ providing in particular compactness results for uniformly finite energy maps $u$ in $Ω$ connecting two wells of $W$ as $x_1\to\pm\infty$.

math.AP↗

arXiv2017

Uniform boundedness principles for Sobolev maps into manifolds

Given a connected Riemannian manifold $\mathcal{N}$, an $m$--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space $W^{s,p}(\mathcal{M}, \mathcal{N})$ imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach--Steinhaus uniform boundedness principle in linear Banach spaces.

math.FA↗

arXiv2017

Metric methods for heteroclinic connections in infinite dimensional spaces

We consider the minimal action problem min \int\_R 1/2 |$γ$'|^2 + W($γ$) dt among curves lying in a non-locally-compact metric space and connecting two given zeros of W $\ge$ 0. For this problem, the optimal curves are usually called heteroclinic connections. We reduce it, following a standard method, to a geodesic problem of the form min \int\_0^1 K($γ$)|$γ$'| dt with K = (2W)^(1/2). We then prove existence of curves minimizing this new action under some suitable compactness assumptions on K, which are minimal. The method allows to solve some PDE problems in unbounded domains, in particular in two variables x, y, when y = t and when the metric space is an L^2 space in the first variable x, and the potential W includes a Dirichlet energy in the same variable. We then apply this technique to the problem of connecting, in a functional space, two different heteroclinic connections between two points of the Euclidean space, as it was previously studied by Alama-Bronsard-Gui and by Schatzman more than fifteen years ago. With a very different technique, we are able to recover the same results, and to weaken some assumptions.

math.AP↗

arXiv2016

Metric methods for heteroclinic connections

We consider the problem $\min\int_{\mathbb{R}} \frac{1}{2}|\dotγ|^2+W(γ)\mathop{}\mathopen{}\mathrm{d} t $ among curves connecting two given wells of $W\geq 0$ and we reduce it, following a standard method, to a geodesic problem of the form $\min\int_0^1 K(γ)|\dotγ|\mathop{}\mathopen{}\mathrm{d} t$ with $K=\sqrt{2W}$. We then prove existence of curves minimizing this new action just by proving that the distance induced by $K$ is proper (i.e. its closed balls are compact). The assumptions on $W$ are minimal, and the method seems robust enough to be applied in the future to some PDE problems.

math.MG↗

arXiv2015

A necessary condition for lower semicontinuity of line energies

We are interested in some energy functionals concentrated on the discontinuity lines of divergence-free 2D vector fields valued in the circle $\mathbb{S}^1$. This kind of energy has been introduced first by P. Aviles and Y. Giga. They show in particular that, with the cubic cost function $f(t)=t^3$, this energy is lower semicontinuous. In this paper, we construct a counter-example which excludes the lower semicontinuity of line energies for cost functions of the form $t^p$ with $0<p<1$. We also show that, in this case, the viscosity solution corresponding to a certain convex domain is not a minimizer.

math.AP↗

arXiv2015

Uniform estimates for a Modica-Mortola type approximation of branched transportation

Models for branched networks are often expressed as the minimization of an energy $M^α$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We study a Modica-Mortola type approximation $M^α_\varepsilon$, introduced by Edouard Oudet and Filippo Santambrogio, which is defined over $H^1$ vector measures. These energies induce some pseudo-distances between $L^2$ functions obtained through the minimization problem $\min \{M^α_\varepsilon (u)\;:\;\nabla\cdot u=f^+-f^-\}$. We prove some uniform estimates on these pseudo-distances which allow us to establish a $Γ$-convergence result for these energies with a divergence constraint.

math.AP↗