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Ginzburg-Landau equation with magnetic effect: non-simply-connected domains. (English) Zbl 0912.58011
The authors consider the Ginzburg-Landau energy functional ${\mathcal H}_\lambda (\Phi,A) =\int_\Omega \left({1\over 2} \bigl| (\nabla-iA)\Phi \bigr |^2+ {\lambda \over 4} \bigl(1-| \Phi|^2 \bigr)^2\right) dx+\int_{\mathbb{R}^3}{1\over 2} | \text{rot} A |^2dx$ which appears as a model of low-temperature superconductivity phenomena. Here $$\Omega\subset\mathbb{R}^3$$ is a bounded domain, $$\Phi$$ is a $$\mathbb{C}$$-valued function in $$\Omega$$ and $$A$$ is a $$\mathbb{R}^3$$-valued function in $$\mathbb{R}^3$$. They consider a minimizer $$(\Phi,A)$$, which physically corresponds to the permanent current of electrons in the material without any outer magnetic field. They prove that if $$\Omega$$ is not simply-connected, there exist many local minimizers provided that $$\lambda$$ is large.

##### MSC:
 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 82D55 Statistical mechanical studies of superconductors 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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