Asymptotic behaviour for minimizers of a Ginzburg-Landau-type functional. (Q2759782)

Let \(G\subset\mathbb R^2\) denote a bounded, simply connected domain with smooth boundary \(\partial G\) and consider a smooth map \(g: \partial G\to{\mathcal S}^1=\{x\in\mathbb R^2 : |x|=1\}\). The authors discuss the functional \((p>1)\) NEWLINE\[NEWLINEE_\epsilon(u,G)={1\over p}\int_G|\nabla u|^p\,dx+{1\over4\epsilon^p}\int_G(1-|u|^2)^2\, dx\quad(\epsilon>0)NEWLINE\]NEWLINE on the class \(W_g=\{v\in W^{1,p}(G,\mathbb R^2) : v|_{\partial G}=g\}\) and discuss the asymptotic behaviour of the minimizers \(u_\epsilon\) as \(\epsilon\searrow0\), for example, convergence towards the \(p\)-harmonic map with boundary values \(g\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].