论文标题
Ginzburg-Landau在平面域上的谐波图放到一般紧凑的真空歧管上
Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold
论文作者
论文摘要
我们研究了一个小参数$ \ varepsilon $倾向于将渐近性行为归为零，是银堡-landau型能量的最小化器，其非线性罚款潜在地消失在紧凑型submanifold $ \ mathcal $ \ mathcal {n} $上，并带有给定的$ \ mathcal {n n} n} $ {n} $ diricled diricled diricled diricled diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet diriclet。我们表明，最小化器收敛到一个单一的$ \ naturcal {n} $ - 有价值的谐波图，在围绕能量的有限点之外，该点在该点之外平稳，并且能量集中并且其位置最小化的能量最小化，使贝特埃尔（Bethuel），brezis and brezis andhéleinfor Circle for Circle $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ s $ $ \ s^1我们还获得了$γ$ -Convergence的结果和统一的Marcinkiewicz弱$ l^2 $或Lorentz $ L^2 $估计的衍生物。我们证明，对相应的Euler-Lagrange方程的解决方案均匀地收敛到约束，并收敛到远离奇点的谐波图。
We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and Hélein for the circle $\mathbb{S}^1$. We also obtain $Γ$-convergence results and uniform Marcinkiewicz weak $L^2$ or Lorentz $L^2$ estimates on the derivatives. We prove that solutions to the corresponding Euler-Lagrange equation converge uniformly to the constraint and converge to harmonic maps away from singularities.
