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Vincent Millot

Publications and source records attributed to Vincent Millot.

At least 19 recordsLinked to original sources
arXiv2025

Gamma-convergence of nonlocal energies for partitions

We prove that certain nonlocal functionals defined on partitions made of measurable sets Gamma-converge to a local functional modeled on the perimeter in the sense of De Giorgi. Those nonlocal functionals involve generalized surface tension coefficients, and are lower semicontinuous even if the coefficients do not satisfy the triangular inequality. It implies a relaxation process in the limit, and provides a novel effect compare to the known gamma-convergence of the fractional perimeter towards the standard perimeter.

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arXiv2025

Fractional multi-phase transitions and nonlocal minimal partitions

This article is devoted to the study of certain models for phase transitions involving nonlocal energies. A first part is concerned with to the asymptotic analysis of a system of fractional elliptic equations of Allen-Cahn type as a characteristic small parameter tends to zero. It is shown that solutions converge to critical points of a nonlocal geometric energy defined over a class of partitions of the domain. A regularity analysis for solutions of the geometric problem is also performed, in the minimizing and non minimizing case. The limiting geometric problem involves generalized surface tension coefficients which might not satisfy the usual triangular inequality. A more detailed regularity analysis for minimizers is performed for 3-partitions, in particular in the case where one triangular inequality strictly holds in the reverse sense.

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arXiv2022

On the Convergence of critical points of the Ambrosio-Tortorelli functional

This work is devoted to study the asymptotic behavior of critical points $\{(u_\varepsilon,v_\varepsilon)\}_{\varepsilon>0}$ of the Ambrosio-Tortorelli functional. Under a uniform energy bound assumption, the usual $Γ$-convergence theory ensures that $(u_\varepsilon,v_\varepsilon)$ converges in the $L^2$-sense to some $(u_*,1)$ as $\varepsilon\to 0$, where $u_*$ is a special function of bounded variation. Assuming further the Ambrosio-Tortorelli energy of $(u_\varepsilon,v_\varepsilon)$ to converge to the Mumford-Shah energy of $u_*$, the later is shown to be a critical point with respect to inner variations of the Mumford-Shah functional. As a byproduct, the second inner variation is also shown to pass to the limit. To establish these convergence results, interior ($\mathscr{C}^\infty$) regularity and boundary regularity for Dirichlet boundary conditions are first obtained for a fixed parameter $\varepsilon>0$. The asymptotic analysis is then performed by means of varifold theory in the spirit of scalar phase transition problems.

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arXiv2022

Torus-like solutions for the Landau-de Gennes model. Part III: torus vs split minimizers

We study the behaviour of global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axially symmetric domains domains diffeomorphic to a ball (a nematic droplet) and in a restricted class of $\bbS^1$-equivariant configurations. It is known from our previous paper \cite{DMP2} that, assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a physically relevant norm constraint in the interior (Lyuksyutov constraint), minimizing configurations are either of \emph{torus} or of \emph{split} type. Here, starting from a nematic droplet with the homeotropic boundary condition, we show how singular (split) solutions or smooth (torus) solutions (or even both) for the Euler-Lagrange equations do appear as energy minimizers by suitably deforming either the domain or the boundary data. As a consequence, when minimizers among $\bbS^1$-equivariant configurations are singular we derive symmetry breaking result for the minimization among all competitors.

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arXiv2022

Isoperimetry and stability properties of balls with respect to nonlocal energies

We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

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arXiv2021

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of $\mathbb{S}^1$-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional axisymmetric domains and in a restricted class of $\mathbb{S}^1$-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing $\mathbb{S}^1$-equivariant (e.g. homeotropic) Dirichlet boundary conditions and a physically relevant norm constraint (Lyuksyutov constraint) in the interior. Relying on results in \cite{DMP1} in the nonsymmetric setting, we prove partial regularity of minimizers away from a possible finite set of interior singularities lying on the symmetry axis. For a suitable class of domains and boundary data we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite union of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit $\mathbb{S}^1$-equivariant harmonic maps into $\mathbb{S}^4$, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

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arXiv2020

Torus-like solutions for the Landau-de Gennes model. Part I: the Lyuksyutov regime

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains. Assuming smooth and uniaxial (e.g. homeotropic) boundary conditions and a corresponding physically relevant norm constraint (Lyuksyutov constraint) in the interior, we prove full regularity up to the boundary for the minimizers. As a consequence, in a relevant range (which we call the Lyuksyutov regime) of parameters of the model we show that even without the norm constraint isotropic melting is anyway avoided in the energy minimizing configurations. Finally, we describe a class of boundary data including radial anchoring which yield in both the previous situations as minimizers smooth configurations whose level sets of the biaxiality carry nontrivial topology. Results in this paper will be largely employed and refined in the next papers of our series. In particular, in [DMP2], we will prove that for smooth minimizers in a restricted class of axially symmetric configurations, the level sets of the biaxiality are generically finite union of tori of revolution.

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arXiv2020

Partial regularity for fractional harmonic maps into spheres

This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.

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arXiv2019

Minimizing 1/2-harmonic maps into spheres

In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $m\geq 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $\mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $\mathbb{S}^1$ has a topological charge equal to $\pm1$.

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arXiv2017

A Ginzburg-Landau model with topologically induced free discontinuities

We study a variational model which combines features of the Ginzburg-Landau model in 2D and of the Mumford-Shah functional. As in the classical Ginzburg-Landau theory, a prescribed number of point vortices appear in the small energy regime; the model allows for discontinuities, and the energy penalizes their length. The novel phenomenon here is that the vortices have a fractional degree $1/m$ with $m\geq 2$ prescribed. Those vortices must be connected by line discontinuities to form clusters of total integer degrees. The vortices and line discontinuities are therefore coupled through a topological constraint. As in the Ginzburg-Landau model, the energy is parameterized by a small length scale $\varepsilon>0$. We perform a complete $Γ$-convergence analysis of the model as $\varepsilon\downarrow0$ in the small energy regime. We then study the structure of minimizers of the limit problem. In particular, we show that the line discontinuities of a minimizer solve a variant of the Steiner problem. We finally prove that for small $\varepsilon>0$, the minimizers of the original problem have the same structure away from the limiting vortices.

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arXiv2016

On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers

In this article, we consider and analyse a small variant of a functional originally introduced in \cite{BLS,LS} to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $\varepsilon>0$ and resembles the (scalar) Ginzburg-Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as $\varepsilon\to0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.

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arXiv2016

Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces

This article is mainly devoted to the asymptotic analysis of a fractional version of the (elliptic) Allen-Cahn equation in a bounded domain $Ω\subset\mathbb{R}^n$, with or without a source term in the right hand side of the equation (commonly called chemical potential). Compare to the usual Allen-Cahn equation, the Laplace operator is here replaced by the fractional Laplacian $(-Δ)^s$ with $s\in(0,1/2)$, as defined in Fourier space. In the singular limit $\varepsilon\to 0$, we show that arbitrary solutions with uniformly bounded energy converge both in the energetic and geometric sense to surfaces of prescribed nonlocal mean curvature in $Ω$ whenever the chemical potential remains bounded in suitable Sobolev spaces. With no chemical potential, the notion of surface of prescribed nonlocal mean curvature reduces to the stationary version of the nonlocal minimal surfaces introduced by L.A. Caffarelli, J.M. Roquejoffre, and O. Savin. Under the same Sobolev regularity assumption on the chemical potential, we also prove that surfaces of prescribed nonlocal mean curvature have a Minkowski codimension equal to one, and that the associated sets have a locally finite fractional $2s^\prime$-perimeter in $Ω$ for every $s^\prime\in(0,1/2)$.

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arXiv2014

On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres

This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. The proof relies on the representation of the square root Laplacian as a Dirichlet-to-Neumann operator in one more dimension, and on the analysis of a boundary version of the Ginzburg-Landau equation. Besides the analysis of the fractional Ginzburg-Landau equation, we also give a general partial regularity result for stationary 1/2-harmonic maps in arbitrary dimension.

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arXiv2013

Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements

Motivated by models of fracture mechanics, this paper is devoted to the analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. In the spirit of gradient flows in metric spaces, such evolutions are defined in terms of curves of maximal unilateral slope, and are constructed by means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah regime is also carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set.

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arXiv2012

A two-gradient approach for phase transitions in thin films

Motivated by solid-solid phase transitions in elastic thin films, we perform a Gamma-convergence analysis for a singularly perturbed energy describing second order phase transitions in a domain of vanishing thickness. Under a two-wells assumption, we derive a sharp interface model with an interfacial energy depending on the asymptotic ratio between the characteristic length scale of the phase transition and the thickness of the film. In each case, the interfacial energy is determined by an explicit optimal profile problem. This asymptotic problem entails a nontrivial dependance on the thickness direction when the phase transition is created at the same rate as the thin film, while it shows a separation of scales if the thin film is created at a faster rate than the phase transition. The last regime, when the phase transition is created at a faster rate than the thin film, is more involved. Depending on growth conditions of the potential and the compatibility of the two phases, we either obtain a sharp interface model with scale separation, or a trivial situation driven by rigidity effects.

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arXiv2009

Gamma-convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of two-dimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity, as the Ginzburg-Landau parameter epsilon tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that, suitably normalized, the energy functionals Gamma-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

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