arXiv subjects

Xavier Lamy

Publications and source records attributed to Xavier Lamy.

48 records · Page 3Linked to original sources

arXiv2017

On the regularity of weak solutions to Burgers equation with finite entropy production

Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a rectifiable jump set of dimension one can be identified, outside which $u$ has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for $BV$ solutions. In the present article we show that the set of non-Lebesgue points of $u$ has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely $μ=\partial_t (u^2/2)+\partial_x(u^3/3)$. We prove Hölder regularity at points where $μ$ has finite $(1+α)$-dimensional upper density for some $α>0$. The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if $μ_+$ has vanishing 1-dimensional upper density, then $u$ is an entropy solution. We obtain a quantitative version of this statement: if $μ_+$ is small then $u$ is close in $L^1$ to an entropy solution.

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

Minimizers of the Landau-de Gennes energy around a spherical colloid particle

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de Gennes model. The nematic is assumed to occupy the exterior of a ball of radius r_0, satisfy homeotropic weak anchoring at the surface of the colloid, and approach a uniform uniaxial state at infinity. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale r_0< >L. The relationship between the radius and the anchoring strength W is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a "Saturn ring" defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen-Frank energy, and a dipole configuration with exactly one point defect is obtained.

math.AP↗

arXiv2014

Vortex structure in p-wave superconductors

We study vortices in p-wave superconductors in a Ginzburg-Landau setting. The state of the superconductor is described by a pair of complex wave functions, and the p-wave symmetric energy functional couples these in both the kinetic (gradient) and potential energy terms, giving rise to systems of partial differential equations which are nonlinear and coupled in their second derivative terms. We prove the existence of energy minimizing solutions in bounded domains $Ω\subset\mathbb R^2$, and consider the existence and qualitative properties (such as the asymptotic behavior) of equivariant solutions defined in all of $\mathbb R^2$. The coupling of the equations at highest order changes the nature of the solutions, and many of the usual properties of classical Ginzburg-Landau vortices either do not hold for the p-wave solutions or are not immediately evident.

math.AP↗

arXiv2014

Biaxial escape in nematics at low temperature

In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain $Ω\subset \mathbb{R}^3$. We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and Hélein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in $Ω$. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class.

math.AP↗

arXiv2014

Persistence of superconductivity in thin shells beyond $H_{c1}$

In Ginzburg-Landau theory, a strong magnetic field is responsible for the breakdown of superconductivity. This work is concerned with the identification of the region where superconductivity persists, in a thin shell superconductor modeled by a compact surface $\mathcal M\subset\mathbb R^3$, as the intensity $h$ of the external magnetic field is raised above $H_{c1}$. Using a mean field reduction approach devised by Sandier and Serfaty as the Ginzburg-Landau parameter $κ$ goes to infinity, we are led to studying a two-sided obstacle problem. We show that superconductivity survives in a neighborhood of size $(H_{c1}/h)^{1/3}$ of the zero locus of the normal component $H$ of the field. We also describe intermediate regimes, focusing first on a symmetric model problem. In the general case, we prove that a striking phenomenon we call freezing of the boundary takes place: one component of the superconductivity region is insensitive to small changes in the field.

math.AP↗

arXiv2014

Characterization of function spaces via low regularity mollifiers

Smoothness of a function $f:{\mathbb R}^n\to {\mathbb R}$ can be measured in terms of the rate of convergence of $f\astρ_\varepsilon$ to $f$, where $ρ$ is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that $ρ$ is adapted to a given scale of spaces. Finally, we examine in detail the case where $ρ$ is a characteristic function.

math.FA↗

arXiv2014

Uniaxial symmetry in nematic liquid crystals

Within the Landau-de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. We show that the uniaxial symmetry constraint is very restrictive and can in general not be satisfied, except in very symmetric situations. For one- and two-dimensional configurations, we characterize completely the uniaxial equilibria: they must have constant director. In the three dimensional case we focus on the model problem of a spherical droplet with radial anchoring, and show that any uniaxial equilibrium must be spherically symmetric. It was known before that uniaxiality can sometimes be broken by energy minimizers. Our results shed a new light on this phenomenon: we prove here that in one or two dimensions uniaxial symmetry is always broken, unless the director is constant. Moreover, our results concern all equilibrium configurations, and not merely energy minimizers.

math.AP↗

arXiv2013

Bifurcation analysis in a frustrated nematic cell

Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behaviour of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture agrees with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and J.R. Kelly. Some of the methods that we use in the present paper apply to other situations, and we present the proofs in a general setting. More precisely, the paper contains the proof of a general uniqueness result for a class of perturbed quasilinear elliptic systems, and general considerations about symmetric solutions and their stability, in the spirit of Palais' Principle of Symmetric Criticality.

math.AP↗

arXiv2013

Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

Let $Ω$ be a smooth bounded simply connected domain in $\mathbb{R}^2$. We investigate the existence of critical points of the energy $E_\varepsilon (u)=1/2\int_Ω|\nabla u|^2+1/(4\varepsilon^2)\int_Ω(1-|u|^2)^2$, where the complex map $u$ has modulus one and prescribed degree $d$ on the boundary. Under suitable nondegeneracy assumptions on $Ω$, we prove existence of critical points for small $\varepsilon$. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in "most" of the domains.

math.AP↗

arXiv2013

A new light on the breaking of uniaxial symmetry in nematics

Within the Landau-de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. For an arbitrary form of the bulk energy density, we show that energy minimizers among uniaxially symmetric configurations are usually not equilibrium configurations. It was known before that uniaxiality can sometimes be broken by energy minimizers. We prove here that this is always the case in one or two dimensions. Even more we prove that equilibrium configurations (not only minimizers) cannot be uniaxial in one or two dimensions (unless they are constant). Namely, uniaxial equilibrium configurations with at least one translational invariance direction must have a uniform director field. In the case of a hybrid nematic cell, or of a capillary with radial anchoring, our result implies that biaxial escape always occurs, and that it does so not only in the core of a defect: uniaxial order is destroyed in the whole cell.

cond-mat.soft↗

arXiv2012

Some properties of the nematic radial hedgehog in Landau-de Gennes' theory

In the Landau-de Gennes theoretical framework of a $Q -tensor description of nematic liquid crystals, we consider a radial hedgehog defect with strong anchoring conditions in a ball $B \subset \mathbb{R}^3$ . We show that the scalar order parameter is monotonic, and we prove uniqueness of the minimizing hedgehog below the spinodal temperature $T^\ast$ .

math.AP↗