Searcharxiv⌕ Search

arXiv subjects

Luc Nguyen

Publications and source records attributed to Luc Nguyen.

51 records · Page 3Linked to original sources
arXiv2015

Instability of point defects in a two-dimensional nematic liquid crystal model

We study a class of symmetric critical points in a variational $2D$ Landau - de Gennes model where the state of nematic liquid crystals is described by symmetric traceless $3\times 3$ matrices. These critical points play the role of topological point defects carrying a degree $\frac k 2$ for a nonzero integer $k$. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when $k\neq \pm 1, 0$.

math.AP↗
arXiv2014

Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals

We study a singular nonlinear ordinary differential equation on intervals $[0,R)$ with $R\le +\infty$, motivated by the Ginzburg-Landau models in superconductivity and Landau-de Gennes models in liquid crystals. We prove existence and uniqueness of positive solutions under general assumptions on the nonlinearity. Further uniqueness results for sign-changing solutions are obtained for a physically relevant class of nonlinearities. Moreover, we prove a number of fine qualitative properties of the solution that are important for the study of energetic stability.

math.AP↗
arXiv2014

Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals

We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (so called "melting hedgehog") in the framework of the Landau - de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary $Q$-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.

math.AP↗
arXiv2013

A response to an article of Xu-Jia Wang

This is a response to the article arXiv:1212.3130v1 by Xu-Jia Wang, where he attempted to address a mathematical question we raised. We point out that, and explain why, the article is far from answering our objections. Moreover, we have more recently found more serious trouble in the paper under discussion based on the false assertion there that minimal radial functions of superharmonic functions (with respect to a conformal Laplacian of a Riemannian metric) are superharmonic.

math.AP↗
arXiv2012

A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone $Γ$ satisfies $μ^+_Γ\le 1$, which includes the $σ_k-$Yamabe problem for $k$ not smaller than half of the dimension of the manifold.

math.AP↗
arXiv2012

A generalized mass involving higher order symmetric function of the curvature tensor

We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the $σ_k$ curvature vanishes at infinity. In addition, with the above assumptions, if the mass is zero, then, near infinity, the manifold is isometric to a Euclidean end.

gr-qc↗
arXiv2011

Existence of singularities in two-Kerr black holes

We show that the angular momentum - area inequality 8π|J| =< A for weakly stable minimal surfaces would apply to (I^+)-regular many-Kerr solutions, if any existed. Hence we remove the undesirable hypothesis in the Hennig-Neugebauer proof of non-existence of well behaved two-component solutions.

gr-qc↗
arXiv2011

Interior Regularity Estimates in High Conductivity Homogenization and Application

In this paper, uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by a epsilon-periodic lattice of highly conducting thin rods. The estimates are derived only at a distance epsilon^{1+tau} (for some tau>0) away from the fibres. This distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in L^p as soon as p>2. One key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a Fourier series expansion with respect to the fibres direction, and weighted by the high-contrast conductivity. The dependence on powers of epsilon of these two-dimensional estimates is shown to be sharp. The initial motivation for this work comes from imaging, and enhanced resolution phenomena observed experimentally in the presence of micro-structures. We use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance epsilon^{1+tau} away from the fibres.

math.AP↗
arXiv2011

A lower bound for the mass of axisymmetric connected black hole data sets

We present a generalisation of the Brill-type proof of positivity of mass for axisymmetric initial data to initial data sets with black hole boundaries. The argument leads to a strictly positive lower bound for the mass of simply connected, connected axisymmetric black hole data sets in terms of the mass of a reference Schwarzschild metric.

gr-qc↗
arXiv2010

Refined approximation for a class of Landau-de Gennes energy minimizers

We study a class of Landau-de Gennes energy functionals in the asymptotic regime of small elastic constant $L>0$. We revisit and sharpen the results in [18] on the convergence to the limit Oseen-Frank functional. We examine how the Landau-de Gennes global minimizers are approximated by the Oseen-Frank ones by determining the first order term in their asymptotic expansion as $L\to 0$. We identify the appropriate functional setting in which the asymptotic expansion holds, the sharp rate of convergence to the limit and determine the equation for the first order term. We find that the equation has a ``normal component'' given by an algebraic relation and a ``tangential component'' given by a linear system.

math.AP↗