论文标题
量化和非量化的能量用于高维的金茨堡 - 兰道涡流
Quantization and non-quantization of energy for higher-dimensional Ginzburg-Landau vortices
论文作者
论文摘要
给定关键点的家族$u_ε：m^n \ to \ mathbb {c} $ for complex ginzburg-landau-landau Energies \ begin {align*} ＆e_ε（u）= \ int_ {m} \ left（\ frac {| du |^2} {2}+\ frac {（1- | u |^2）^2} {4ε^2} \ right），\ end offerold $ m $ $ $ $ e___________________________________________________涡度设置$ \ {|u_ε| \ leq \ frac {1} {2} {2} \} $随后收敛于支撑固定的，可纠正的$（n-2）$ - varifold $ v $在内部的内部，其特征是限制的限制$ \ lim_}} ^ 0}的集中部分。 \ frac {e_ε（u_ε）} {π| \logε| } $的归一化能量度量。 当$ n = 2 $或解决方案$u_ε$是能量最小化时，众所周知，此varifold $ v $是积分的；即$（n-2）$ - 密度$θ_{n-2}（| v |，x）$ | v | $ of $ \ mathbb {n} $ at $ | v | $ -a.e。 $ x \ m $。在本文中，我们表明，对于$e_ε（u_ε）= o（| \logε|）$的一般临界点，在尺寸$ n \ geq 3 $中，这种能量量化现象只能保持密度小于$ 2 $：不小于$ 2 $：不，即，我们证明了密度$θ_{n-2} n-2} nimimit limip v | v | v | v | v | v | v | v | v |，x） $ \ {1 \} \ cup [2，\ infty）$ at $ | v | $ -a.e。 $x\in M$, and show that this is sharp, in the sense that for any $n\geq 3$ and $θ\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_ε$ for $E_ε$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $θ$.
Given a family of critical points $u_ε:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_ε(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4ε^2}\right), \end{align*} on a manifold $M$, with natural energy growth $E_ε(u_ε)=O(|\logε| )$, it is known that the vorticity sets $\{|u_ε|\leq \frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{ε\to 0} \frac{e_ε(u_ε)}{π|\logε| }$ of the normalized energy measures. When $n=2$ or the solutions $u_ε$ are energy-minimizing, it is known moreover that this varifold $V$ is integral; i.e., the $(n-2)$-density $Θ_{n-2}(|V|,x)$ of $|V|$ takes values in $\mathbb{N}$ at $|V|$-a.e. $x\in M$. In the present paper, we show that for a general family of critical points with $E_ε(u_ε)=O(|\logε| )$ in dimension $n\geq 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density $Θ_{n-2}(|V|,x)$ of the limit varifold takes values in $\{1\}\cup [2,\infty)$ at $|V|$-a.e. $x\in M$, and show that this is sharp, in the sense that for any $n\geq 3$ and $θ\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_ε$ for $E_ε$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $θ$.
