• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.1019-1043
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• 1998

In this paper we study some asymptotics for solutions of the Ginzburg-Landau equations with Dirichlet boundary conditions. We consider the solutions ( $u_{\in}$, $A_{\in}$) which minimize the Ginzburg-Landau energy functional $E_{\in}$(u, A). We show that the solutions ( $u_{\in$}$ , $A_{\in}$) converge to some ( $u_{*}$, $A_{*}$) in various norms as the coupling parameter $\in$longrightarrow0.ow0.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.585-603
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• 2007

The author studies the local $H\ddot{o}lder$ convergence of the solution to an evolution equation of p-Ginzburg-Landau type, to the heat flow of the p-harmonic map, when the parameter tends to zero. The convergence is derived by establishing a uniform gradient estimation for the solution of the regularized equation.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.139-165
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• 2020

Mirror symmetry for del Pezzo surfaces was studied in [3] where they suggested that the mirror should take the form of a Landau-Ginzburg model with a particular type of elliptic fibration. This argument came from symplectic considerations of the derived categories involved. This problem was then considered again but from an algebro-geometric perspective by Gross, Hacking and Keel in [8]. Their construction allows one to construct a formal mirror family to a pair (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. In the case where the self intersection matrix for D is not negative semi-definite it was shown in [8] that this family may be lifted to an algebraic family over an affine base. In this paper we perform this construction for all smooth del Pezzo surfaces of degree at least two and obtain explicit equations for the mirror families and present the mirror to dP₂ as a double cover of ℙ².

• Progress in Superconductivity
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• v.11 no.2
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• pp.112-117
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• 2010

We examine phase transition of the spin density wave and $\pi$ phase shifted superconductivity in the Fe pnictide superconductors. The phase diagram is described in the plane of the temperature T and the doping x with the combination of Ginzburg-Landau expansion of the free energy near the multi-critical temperature $T_c$ and the self-consistent numerical iterations of the gap equations. The phase separation or coexistence is determined by computing the 4-th order terms of the free energy which is confirmed by the numerical calculations. We can show the phase coexistence when the spin density wave is incommensurate. And the first order phase transition is observed near the boundary between commensurate and incommensurate spin density wave.

• Macromolecular Research
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• v.14 no.2
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• pp.155-165
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• 2006

The present article describes the experimental and theoretical observations on the formation of holographic, polymer-dispersed, liquid crystals and electrically switchable, photonic crystals. A phase diagram of the starting mixture of nematic liquid crystal and photo-reactive triacrylate monomer was established by means of differential scanning calorimetry (DSC) and cloud point measurement. Photolithographic patterns were imprinted on the starting mixture of LC/triacrylate via multi-beam interference. A similar study was extended to a dendrimer/photocurative mixture as well as to a single component system (tetra-acrylate). Theoretical modeling and numerical simulation were carried out based on the combination of Flory-Huggins free energy of mixing and Maier-Saupe free energy of nematic ordering. The combined free energy densities were incorporated into the time-dependent Ginzburg-Landau (Model C) equations coupled with the photopolymerization rate equation to elucidate the spatio-temporal structure growth. The 2-D photonic structures thus simulated were consistent with the experimental observations. Furthermore, 3-D simulation was performed to guide the fabrication of assorted photonic crystals under various beam-geometries. Electro-optical performance such as diffraction efficiency was evaluated during the pattern photopolymerization process and also as a function of driving voltage.