×

Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. II. (English) Zbl 0905.49006

[See also the preceding review of Part I].
Let \(G\) be a bounded, simply connected, smooth domain in 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.

MSC:

49J35 Existence of solutions for minimax problems
49K20 Optimality conditions for problems involving partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
55M25 Degree, winding number
58E20 Harmonic maps, etc.

Citations:

Zbl 0905.49005
PDFBibTeX XMLCite
Full Text: DOI