Boundary regularity and the Dirichlet problem for harmonic maps. (English) Zbl 0547.58020

Schoen, Richard; Uhlenbeck, Karen

J. Differ. Geom. 18, 253-268 (1983).

Closely related to their fundamental paper in ibid. 17, 307-335, Correction ibid. 19, 329 (1982; Zbl 0521.58021) the authors prove here boundary regularity for energy minimizing maps with prescribed Dirichlet boundary condition, more precisely, they show that, for compact M with \(C^{2,\alpha}\) boundary, an \(\tilde E\)-minimizing \(u\in L^ 2_ 1(M,N)\) whose image lies a.e. in a compact subset \(N_ 0\) of the range N is of class \(C^{2,\alpha}\) in a full neighbourhood of \(\partial M\) provided that \(u/\partial M\in C^{2,\alpha}(\partial M,N_ 0)\) (in fact, the singular set of u is compact and lies in the interior of M). By the direct method, they also recover an earlier result of J. Sacks and the second author [Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014)] and show harmonic representability of elements of \(\pi_ k(N)\) for certain N. Finally, they settle the approximation problems posed by J. Eells and L. Lemaire [Reg. Conf. Ser. Math. 50 (1983; Zbl 0515.58011)] by showing that \(L^ 2_ 1\)-maps from 2- manifolds can be approximated by smooth maps and giving an example of an \(L^ 2_ 1\)-map from the 3-ball to the 2-sphere which is not an \(L^ 2_ 1\)-limit of continuous maps.

Reviewer: G.Toth

MSC:

58E20

Harmonic maps, etc.

Keywords:

harmonic map; Dirichlet problem; regularity

Citations:

Zbl 0521.58021; Zbl 0462.58014; Zbl 0515.58011