arXiv subjects

Rémy Rodiac

Publications and source records attributed to Rémy Rodiac.

29 records · Page 2Linked to original sources

arXiv2019

Metric characterization of the sum of fractional Sobolev spaces

We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for $\ell \in \mathbb{N}^*$, $s_i\in (0, 1)$ and $p_i \in [1, +\infty)$, $u : Ω\to \mathbb{R}$ can be decomposed as $u = u_1+\dotsc+u_\ell$ with $u_i \in \dot{W}^{s_i,p_i}(Ω)$ if and only if $$ \iint\limits_{Ω\times Ω} \min_{1 \le i \le \ell} \frac{|u (x) - u (y)|^{p_i}}{|x - y|^{n+s_ip_i}}\,\mathrm{d}x \,\mathrm{d}y <+\infty. $$

math.AP↗

arXiv2019

Critical weak-$L^{p}$ differentiability of singular integrals

We establish that for every function $u \in L^1_\mathrm{loc}(Ω)$ whose distributional Laplacian $Δu$ is a signed Borel measure in an open set $Ω$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $Ω$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $Δu$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = α\}$ and $\{\nabla u = e\}$, for every $α\in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.

math.FA↗

arXiv2018

Description of limiting vorticities for the magnetic 2D Ginzburg-Landau equations

Let $Ω$ be a bounded open set in $\mathbb{R}^2$. The aim of this article is to describe the functions $h$ in $H^1(Ω)$ and the Radon measures $μ$ which satisfy $-Δh+h=μ$ and $ div(T_h)=0$ in $Ω$, where $T_h$ is a $2\times 2$ matrix given by $(T_h)_{ij}=2\partial_ih\partial_jh- (|\nabla h|^2+h^2)δ_{ij}$ for $i,j=1,2$. These equations arise as equilibrium conditions satisfied by limiting vorticities and limiting induced magnetic fields of solutions of the magnetic Ginzburg-Landau equations as shown by Sandier-Serfaty. Let us recall that they obtained that $|\nabla h|$ is continuous in $Ω$. We prove that if $x_0$ in $ Ω$ belongs to $\supp μ$ and is such that $|\nabla h(x_0)|\neq 0$ then $μ$ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a $\mathcal{C}^1$-curve near $x_0$ whereas $μ_{\lfloor \{| \nabla h|=0\}}=h_{|\{ |\nabla h|=0\}}$. We also prove that if $Ω$ is smooth bounded and star-shaped and if $h=0$ on $\partial Ω$ then $h \equiv 0$ in $Ω$. This rules out the possibility of having critical points of the Ginzburg-Landau energy with a number of vortices much larger than the applied magnetic field $h_{ex}$ in that case.

math.AP↗

arXiv2017

One dimensional phase transition problem modelling striped spin orbit coupled Bose-Einstein condensates

We study the behaviour of a Modica-Mortola phase transition type problem with a non-homogeneous Neumann boundary condition. According to the parameters of the problem, this leads to the existence of either one component occupying most of the condensate with an outer boundary layer containing the other component, or to many interfaces, on a periodic pattern. This is related to the striped behaviour of a two component Bose-Einstein condensate with spin orbit coupling in one dimension. We prove that minimizers of the full Gross-Pitaevskii energy in 1D converge, in the Thomas-Fermi limit of strong intra-component interaction, to those of the simplified Modica-Mortola problem we have studied in the first part.

math.AP↗

arXiv2017

On the convergence of minimizers of singular perturbation functionals

The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions $n\geq 3$, and to general nonlinearities. Our proof follows a well-known scheme, relying on small energy estimate and monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular this extends existing results in 3-dimensional settings. In higher dimensions we also deal with additional difficulties concerning the boundary monotonicity formula.

math.AP↗

arXiv2017

Variational reduction for semi-stiff Ginzburg-Landau vortices

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^2$. For $ε>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-Δu=\frac{1}{ε^2}(1-|u|^2)u$ in $Ω$ such that on $\partial Ω$ u satisfies $|u|=1$ and $u\wedge \partial_νu=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions we use a variational reduction method very similar to the one used by del Pino-Kowalczyk-Musso to find solutions of the Ginzburg-Landau equations with Dirichlet and homogeneous Neumann boundary conditions. We obtain the exact same result as the authors of the aforementioned article obtained for the Neumann problem. This is because the renormalized energy for the Neumann problem and for the semi-stiff problem are the same. In particular if $Ω$ is simply connected a solution with degree one on the boundary always exists and if $Ω$ is not simply connected then for any $k\geq 1$ a solution with $k$ vortices of degree one exists.

math.AP↗

arXiv2016

On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting

Let $Ω$ be a smooth bounded axisymmetric set in $\R^3$. In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis, a phenomenon of concentration of energy can occur preventing the strong convergence in $W^{1,2}(Ω,\mathbb{R}^3)$ of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where $Ω$ is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are $W^{1,2}$-quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for $3$D neo-Hookean materials.

math.AP↗

arXiv2015

Boundary regularity of weakly anchored harmonic maps

In this note we study the boundary regularity of minimizers of a family of weak anchoring energies that model the states of liquid crystals. We establish optimal boundary regularity in all dimensions $n\geq 3 .$ In dimension $n=3,$ this yields full regularity at the boundary which stands in sharp contrast with the observation of boundary defects in physics works. We also show that, in the cases of weak and strong anchoring, regularity of minimizers is inherited from that of their corresponding limit problems.The analysis rests in a crucial manner on the fact that the surface and Dirichlet energies scale differently; we take advantage of this fact to reduce the problem to the known regularity of tangent maps with zero Neumann conditions.

math.AP↗

arXiv2015

Regularity properties of stationary harmonic functions whose Laplacian is a Radon measure

We study the regularity of Radon measures $μ$ which satisfy that there exists a function $h_μ$ in $H^1(Ω)$, stationary harmonic such that $Δh_μ=μ$ in $Ω$ (here $Ω$ is an open set of $\mathbb{R}^2$). Such conditions appear in physical contexts such as the study of a limiting vorticity measure associated to a family $(u_\varepsilon)_\varepsilon$ of solutions of the Ginzburg-Landau system without magnetic field. Under these conditions we prove that locally there exists a harmonic function $H$ such that the support of the measure is contained in the set of zeros of $H$. Using the local structure of the set of zeros of harmonic functions we can thus obtain that locally the support of $μ$ is a union of smooth simple

math.AP↗

arXiv2015

Existence and non existence results for minimizers of the Ginzburg-Landau energy with prescribed degrees

Let $\mathcal{D} =Ω\setminus\barω \subset \mathbb{R}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_ε(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4ε^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D} \rightarrow \mathbb{C}$, and look for minimizers of $E_ε$ with prescribed degrees $deg(u,\partial Ω)=p$, $deg(u,\partial ω)=q$ on the boundaries of the domain. For large $ε$ and for balanced degrees, i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also prove non-existence of minimizers of $E_ε$, for large $ε$, in the case $p\neq q$, $pq>0$ and $\mathcal{D}$ is a circular annulus with large capacity (corresponding to "thin" annulus). Our approach relies on similar results obtained for the Dirichlet energy $E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$, the existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.

math.AP↗

arXiv2015

Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids

Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components of $\partial A$. This variational problem is a problem with lack of compactness hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf differential and on a strong link between this problem and the problem of finding all minimal surfaces bounded by two $p$ covering of circles in parallel planes. We then construct new immersed minimal surfaces in $\mathbb{R} ^3$ with this property. These surfaces are obtained by bifurcation from a family of $p$-coverings of catenoids.

math.AP↗