Let $\Omega$ be a smooth bounded domain in $\mathbb R^2$. For $\varepsilon>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $$ -\Delta u=\frac{1}{\varepsilon^2}(1-|u|^2)u \ \text{ in $\Omega$} $$ such that on $\partial \Omega$ u satisfies $|u|=1$ and $u\wedge \partial_\nu u=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions, we use a variational reduction method very similar to the one used in [12]. We obtain the exact same result as the authors of the aforementioned article obtained for the Neumann problem. This is because the renormalized energy for the Neumann problem and for the semi-stiff problem are the same. In particular, if $\Omega$ is simply connected a solution with degree one on the boundary always exists and if $\Omega$ is not simply connected, then for any $k\geq 1$ a solution with $k$ vortices of degree one exists.

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半刚性金茨堡-朗道涡的变分减小

令$ \ Omega $为$ \ mathbb R ^ 2 $中的光滑有界域。对于$ \ varepsilon> 0 $小，我们构造了Ginzburg-Landau方程$$-\ Delta u = \ frac {1} {\ varepsilon ^ 2}（1- | u | ^ 2）u \的非常数解。 \ text {in $ \ Omega $} $$，这样在$ \ partial \ Omega $上，u满足$ | u | = 1 $和$ u \ wedge \ partial_ \ nu u = 0 $。这些边界条件称为半刚性，介于Dirichlet和齐次Neumann边界条件之间。为了构造这样的解决方案，我们使用与[12]中使用的方法非常相似的变分减少方法。我们得到的结果与前述文章针对Neumann问题获得的作者完全相同。这是因为Neumann问题和半刚性问题的重新归一化能量相同。尤其是，