# zbMATH — the first resource for mathematics

Boundary regularity and the Dirichlet problem for harmonic maps. (English) Zbl 0547.58020
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:
 5.8e+21 Harmonic maps, etc.
##### Keywords:
harmonic map; Dirichlet problem; regularity
Full Text: