标题： 二维Ginzburg-Landau方程平均场极限涡量的稳定性条件 显示英文标题

标题： Stability conditions for mean-field limiting vorticities of the Ginzburg-Landau equations in 2D

摘要： 我们分析当$\varepsilon$（GL参数的倒数）趋于零时，Ginzburg-Landau(GL)方程稳定解的极限，并且在施加的磁场为$|\log \varepsilon |$的量级而总能量为$|\log \varepsilon|^2$的量级的范围内。为了做到这一点，我们在GL能量的第二个内变分中取极限。主要困难在于理解仅在$H^1$中弱收敛的函数的导数所涉及的二次项的收敛性。我们使用能量收敛的假设，由Sandier-Serfaty通过在第一个内变分中取极限得到的极限临界条件以及极限涡旋性质来找到所有所需二次项的极限。最后我们研究了我们得到的极限稳定性条件。在有磁场的情况下，我们研究了一个在正方形$\Omega=(-L,L)^2$中支持在一条线上的可接受极限涡旋的例子，并证明如果$L$足够小，该涡旋满足极限稳定性条件，而当$L$足够大时，它不再满足该条件。 在没有磁场的情况下，我们使用Iwaniec-Onninen的一个结果来证明，满足一阶极限临界条件的$H^{-1}(\Omega)$中的每个测度也满足二阶极限稳定性条件。 显示英文摘要

摘要： We analyse the limit of stable solutions to the Ginzburg-Landau (GL) equations when $\varepsilon$, the inverse of the GL parameter, goes to zero and in a regime where the applied magnetic field is of order $|\log \varepsilon |$ whereas the total energy is of order $|\log \varepsilon|^2$. In order to do that we pass to the limit in the second inner variation of the GL energy. The main difficulty is to understand the convergence of quadratic terms involving derivatives of functions converging only weakly in $H^1$. We use an assumption of convergence of energies, the limiting criticality conditions obtained by Sandier-Serfaty by passing to the limit in the first inner variation and properties of limiting vorticities to find the limit of all the desired quadratic terms. At last we investigate the limiting stability condition we have obtained. In the case with magnetic field we study an example of an admissible limiting vorticity supported on a line in a square $\Omega=(-L,L)^2$ and show that if $L$ is small enough this vorticiy satisfies the limiting stability condition whereas when $L$ is large enough it stops verifying that condition. In the case without magnetic field we use a result of Iwaniec-Onninen to prove that every measure in $H^{-1}(\Omega)$ satisfying the first order limiting criticality condition also verifies the second order limiting stability condition.