/pdf/convergence-of-the-allen-cahn-equation-to-brakke-s-motion-by-dsnlct0wdw.pdf

Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature

LetΩ ⊂ ℝ2 be a smooth bounded simply connected domain. Consider the functional
 
$$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$

 on the classH

 g
 1
 ={u eH
1(Ω; ℂ);u=g on ∂Ω} whereg:∂Ω∂ → ℂ is a prescribed smooth map with ¦g¦=1 on ∂Ω∂ and deg(g, ∂Ω)=0. Let uu
e be a minimizer for Ee onH

 g
 1
 . We prove that ue → u0 in
 
$$C^{1,\alpha } (\bar \Omega )$$

 as e → 0, where u0 is identified. Moreover
 
$$\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 $$

 .

Asymptotics for the minimization of a Ginzburg-Landau functional

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in $W^{1,2}$, to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer. 
We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.

Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond

Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the constrained problems can be iteratively solved by computing the corresponding unconstrained problems and orthogonality constrained quadratic problems with analytic solutions. As applications, we demonstrate the robustness of our method in several problems including direction fields correction, noisy color image restoration and global conformal mapping for genus-0 surfaces construction. Numerical comparisons with existing methods are also conducted to illustrate the efficiency of our algorithms.

A Splitting Method for Orthogonality Constrained Problems

Lower Bounds for the Energy of Unit Vector Fields and Applications

Limite singulière pour la minimisation de fonctionnelles du type Ginzburg-Landau

On etudie le comportement quand e→0 des solutions (u e ) a valeurs complexes de l'equation de Ginzburg-Landau -Δu e =1/e 2 u e (1-|u e | 2 ) sur Ω⊂R 2 avec condition de Dirichlet u e =g sur ∂Ω. On met en evidence les proprietes particulieres des solutions minimisantes

Tourbillons de Ginzburg-Landau et énergie renormalisée

The theory for finite type solutions developped in Chapter 8 can be generalized in order to represent all harmonic maps from a simply connected surface to symmetric spaces like the sphere S2 This has been developped by J Dorfmeister, F Pedit and HY Wu and leads to a Weierstrass type representation [30]

Weierstrass type representations

A remark on compact H-surfaces into ${\mathbb R}^3$

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