Remarks on approximate harmonic maps (Q1896466)

Given Riemannian manifolds \(M\) and \(N\) with \(N\) embedded in some space \(\mathbb{R}^k\), the Dirichlet energy for mappings \(M\to N\) is approximated by the functional \(\int_M |\nabla u|^2+ {1\over \varepsilon^2} d^2 (u, N) dx\) \((0< \varepsilon< 1)\) defined on spaces of mappings \(u: M\to \mathbb{R}^k\) where \(d( \cdot, N)\) measures the distance to the manifold \(N\). For the special choice \(N= S^1\) and for two-dimensional domains \(M\) the authors prove uniform estimates for minimizers of the above energy in the class \(H^1 (M, \mathbb{C})\) with prescribed boundary data \(g\) such that \(|g(x) |=1\) and \(\deg (g, \partial M) =0\). They apply these results to give a short proof of some theorems due to \textit{F. Bethuel}, \textit{H. Brezis} and \textit{F. Helein} [Calc. Var. Partial Differ. Equ. 1, No. 2, 123-148 (1993)]\ concerning asymptotic limits for the Ginzburg-Landau model of scalar fields. Moreover, a new proof of the Schoen-Uhlenbeck regularity theorem [\textit{R. Schoen} and \textit{K. Uhlenbeck}, J. Differ. Geom. 17, 307-335 (1982; Zbl 0521.58021)]\ is given and it is shown how to apply this theorem to obtain uniqueness for the harmonic map heat flow.