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Limiting behavior of the Ginzburg-Landau functional. (English) Zbl 1028.49015
Authors’ abstract: “We continue our study of the functional \[ \mathbb{E}_\varepsilon (u):=\int_U {1\over 2}|\nabla u|^2+{1\over 4 \varepsilon^2} \bigl(1-|u|^2 \bigr)^2dx, \] for \(u\in H^1(U; \mathbb{R}^2)\), where \(U\) is a bounded, open subset of \(\mathbb{R}^2\). Compactness results for the scaled Jacobian of \(u^\varepsilon\) are proved under the assumption that \(\mathbb{E}_\varepsilon (u^\varepsilon)\) is bounded uniformly by a function of \(\varepsilon\). In addition, the Gamma limit of \(\mathbb{E}_\varepsilon (u^\varepsilon)/ (\ln\varepsilon)^2\) is shown to be \[ \mathbb{E}(v): =\tfrac 12\|v \|^2_2+ \|\nabla \times v\|_{\mathcal M}, \] where \(v\) is the limit of \(j (u^\varepsilon)/ |\ln\varepsilon |\), \(j (u^\varepsilon): =u^\varepsilon \times Du^\varepsilon\), and \(\|\cdot \|_{\mathcal M}\) is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional \[ \mathbb{F}_\varepsilon (u,A;h_{\text{ext}}) :=\int_U{1\over 2}|\nabla_A u|^2+{1\over 4\varepsilon^2} \bigl(1-|u|^2 \bigr)^2+ {1 \over 2}|\nabla\times A-h_{\text{ext}}|dx, \] with external magnetic field \(h_{\text{ext}} \approx H|\ln \varepsilon|\). The Gamma limit of \(\mathbb{F}_\varepsilon/ (\ln\varepsilon)^2\) is calculated to be \[ \mathbb{F}(v,a;H): = \tfrac 12\bigl [\|v-a\|^2_2+ \|\nabla\times v\|_{\mathcal M}+\|\nabla \times a-H\|^2_2 \bigr], \] where \(v\) is as before, and \(a\) is the limit of \(A^\varepsilon/|\ln\varepsilon|\)”.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
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