IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

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Regularity of minima: an invitation to the Dark Side of the Calculus of Variations

We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically well-posed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for gray-scale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this $L^1$ vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticity-brightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities.

https://www.aimsciences.org/article/exportPdf?id=df11435b-8f2e-43bd-8c1c-07f15d595cfd

Fast dual minimization of the vectorial total variation norm and applications to color image processing

Multiple integrals in the calculus of variations and nonlinear elliptic systems

This paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. In dimension two, we establish both interior and boundary regularity theorems for such a flow under smallness conditions. As a consequence, we establish the existence of global (in time) weak solutions on a bounded smooth domain in $${\mathbb{R}^2}$$
 which are smooth everywhere with possible exceptions of finitely many singular times.

https://www.math.purdue.edu/~wang2482/LLF.pdf

Liquid crystal flows in two dimensions

In the 1960s, Ericksen and Leslie established the hydrodynamic theory for modelling liquid crystal flow. In this paper, we investigate a simplified model of the Ericksen–Leslie system, which is a system of the Navier–Stokes equations coupled with the harmonic map flow. We prove global existence of solutions to the Ericksen–Leslie system in $${\mathbb{R}^{2}}$$
 with initial data, where the solutions are regular except for at a finite number of singular times.

Global existence of solutions of the simplified Ericksen–Leslie system in dimension two

We prove a global existence of solutions for the Landau-Lifshitz equation of the ferromagnetic spin chain from am-dimensional manifoldM into the unit sphereS2 of ℝ3 and establish some new links between harmonic maps and the solutions of the Landau-Lifshitz equation.

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps

http://www.ims.cuhk.edu.hk/publications/reports/2010-05.pdf

Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2

In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in ℝ3 for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on the other one; (iv) a new one, which characterizes the maximal existence time of the strong solutions to the Ericksen-Leslie system in terms of Serrin type norms of the strong solutions to the Ginzburg-Landau approximate system. Furthermore, we also prove that the strong solutions of the Ginzburg-Landau approximate system converge to the strong solution of the Ericksen-Leslie system up to the maximal existence time.

Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3

The global existence of weak solutions for the heat ow of p-harmonic maps of closed Riemannian manifolds into spheres is proved in this paper.

Min-Chun Hong

Papers

Global existence of solutions of the simplified Ericksen–Leslie system in dimension two

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps

Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2

Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3

Heat-Flow of P-Harmonic Maps with Values Into Spheres