Asymptotic behavior of the regularized minimizer of an energy functional in higher dimensions. (English) Zbl 1215.49004
Summary: This paper is concerned with the asymptotic behavior of the regularized minimizer \(u_\varepsilon=(u_{\varepsilon 1},u_{\varepsilon 2},\ldots,u_{\varepsilon n+1})\) of an energy functional
\[
E_\varepsilon (u,G)=\frac1 n\int_G|\nabla u|^n\,dx+\frac{1}{2\varepsilon^n}\int_G u^2_{n+1}\, dx
\]
when \(\varepsilon\to 0\), where \(G\subset \mathbb{R}^n\) is a bounded domain. The author proves \(W^{1,n}\)-convergence of minimizers to the map \(u_n=(u_n',0)\), where \(u_n'\) is an \(n\)-harmonic map. In addition, the relation between the zeros of \(u_{\varepsilon 1}^2+u_{\varepsilon 2}^2+\ldots+u_{\varepsilon n}^2\) and the singularities of \(u_n'\) is given qualitatively.
MSC:
| 49J10 | Existence theories for free problems in two or more independent variables |
| 35Q61 | Maxwell equations |
| 82D40 | Statistical mechanics of magnetic materials |
| 58E20 | Harmonic maps, etc. |
Keywords:
asymptotic behavior; \(n\)-harmonic map; \(W^{1,n}\) convergence; Ginzburg-Landau functionalReferences:
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