Asymptotic behavior of regularizable minimizers of a Ginzburg-Landau functional in higher dimensions (Q2703370)

Having a bounded, simply connected, smooth domain \(G\subset \mathbb{R}^n\), the paper considers the functional NEWLINE\[NEWLINEE_\varepsilon(u, G):= p^{-1} \int_G|\nabla u|^p dx+ (4\varepsilon^p)^{-1} \int_G(1-|u|^2)^2 dxNEWLINE\]NEWLINE for \(p> 1\) and \(\varepsilon> 0\) small. Fixing a smooth mapping from the boundary \(\partial G\) into \(S^{n-1}\) satisfying \(\deg(g,\partial G)\neq 0\), the asymptotic behavior for \(\varepsilon\to 0\) and the location of zeroes of the minimizer \(u_\varepsilon\) over the set \(\{v\in W^{1,p}(G, \mathbb{R}^n)\); \(v|_{\partial G}= g\}\) are studied. Attention is payed on the minimizers, called regularizable, that can be attained (for \(\tau\to 0\)) by minimizers of the regularized functional involving \((|\nabla u|^2+ \tau)^{p/2}\) instead of \(|\nabla u|^p\). It is shown that, e.g., some zero of such \(u_\varepsilon\) is near to a singularity of an \(n\)-harmonic map solving \(-\text{div}(|\nabla u|^{n-2}\nabla u)= u|\nabla u|^n\), and that \(u_\varepsilon\) converges (in terms of a suitable subsequence) to \(u_n\) in \(W^{1,n}_{\text{loc}}\)-sense out of singularities of \(u_n\).