Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. II
[See also the preceding review of Part I]. Let \(G\) be a bounded, simply connected, smooth domain in \textbf{R}\(^2\), \(g:\partial G\rightarrow S^1\) a smooth boundary data of topological degree \(d>0\) and \(p:\overline{G}\rightarrow (0,+\infty)\) a smooth function. The authors study several problems related to the behavior of minimizers \(u_\varepsilon\) for the Ginzburg-Landau functional \[ E_\varepsilon (u)={2}^{-1}\int_Gp|\nabla u|^2+{4\varepsilon^2}^{-1} \int_G(1-| u|^2)^2 \] with respect to the class \(H^1_g=\{ u\in H^1\); \(u=g\) on \(\partial G\}\). Define \(p_0:=\min\{ p(x)\); \(x\in\overline {G}\}\) and \(\Lambda :=p^{-1}(p_0) =\Lambda_i\cup\Lambda_b\), where \(\Lambda_i:=\Lambda\cap G\) and \(\Lambda_b:=\Lambda \cap\partial G\). Set \(K=\text{card} \Lambda\), \(K_i=\text{card} \Lambda_i\) and \(K_b=\text{card} \Lambda_b\). The main hypotheses made throughout the paper are: \[ \Lambda =\{ a_1,\dots ,a_K\}\subset G\quad\text{with}\quad K=K_i<d, \] and there exist \(K\) positive-definite quadratic forms \(Q_1,\dots ,Q_K\) such that \[ p(x)=p_0+Q_j(x-a_j)+o(| x-a_j| ^2)\quad\text{in a neighbourhood of \(a_j\)},\;1\leq j\leq K. \] Under these hypotheses the authors prove that \[ E_\varepsilon (u_\varepsilon)=\pi p_0\{ d| \log\varepsilon | +2^{-1}(F(d,k)- d)\log (| \log\varepsilon |)\}+O(1)\quad\text{as} \varepsilon\rightarrow 0, \] where \(F(d,k)=\min\{ \sum_{j=1}^k\delta_j^2; (\delta_1,\dots ,\delta_k)\in (\mathbf{Z}^+)^k,\;\sum_{j=1}^k\delta_j=d\}\). Moreover, the configuration of vortices coincides with \(\{ a_1,\dots ,a_K\}\) and the configuration of corresponding degrees \((d_1,\dots d_K)\) minimizes the functional \(F(d,K)\). The paper also gives a more precise estimate for the term ``\(O(1)\)'' in terms of the notion of renormalized energy, which is due to Bethuel, Brezis and Hélein. The paper develops interesting methods in order to study the contribution of a nonconstant positive weight in the location of the singularities at the limit.
scientific article; zbMATH DE number 1176742