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Harmonic maps with defects. (English) Zbl 0608.58016
The authors study the problem of minimizing energy in certain classes C of maps $$v: {\mathbb{R}}^ 3\supset \Omega \to S^ 2$$ defined on some 3- dimensional region $$\Omega$$ taking values in the two-sphere $$S^ 2$$. In the first case C consists of all functions v which are continuous except for a finite number of points $$x_ 1,...,x_{\ell}$$ in $$\Omega$$ with prescribed degree $$\deg (v,x_ i)=d_ i$$ and finite energy $$E(v):=\int_{\Omega}| Dv|^ 2 dx$$. Then it is shown that $(1)\quad \inf_{v\in C}E(v)=8\pi L,$ where L is a geometric quantity depending on $$\Omega$$, the distances between the points $$x_ i$$ and on the degrees $$d_ i$$. The upper bound follows by constructing an almost minimizer, the proof of the lower bound is more involved and based on the analysis of the so-called D-field associated to $$v\in C$$. Moreover, equation (1) is extended to the case that the point singularities $$x_ i$$ are replaced by ”singular holes” (i.e. disjoint compact subsets of $$\Omega)$$. The second part of the paper deals with the Dirichlet problem for maps $$v: {\mathbb{R}}^ 3\supset \Omega \to S^ 2$$ with prescribed boundary function $$g: \partial \Omega \to S^ 2$$. By a result of R. Schoen and K. Uhlenbeck [J. Differ. Geom. 17, 307-335, correction ibid. 19, 329 (1982; Zbl 0521.58021); 18, 253-268 (1983; Zbl 0547.58020)] a minimizer u is smooth except for a finite number of singular points $$x_ 1,...,x_ k$$. It is shown that $$\deg (u,x_ i)=\pm 1$$ and that (near $$x_ i)$$ u behaves like $$R(x+x_ i)/| x-x_ i|$$ for a rotation R. This result is a consequence of another theorem saying that in case $$\Omega =B^ 3$$ the homogeneous extension g(x/$$| x|)$$ is E minimizing for boundary values g iff g is constant or a rotation. In a final chapter the authors replace $${\mathbb{R}}^ 3$$, $$S^ 2$$, $$\int | Dv|^ 2$$ by $${\mathbb{R}}^ n$$, $$S^{n-1}$$ and $$\int | Dv|^{n-1}$$ for some $$n\geq 3$$ and prove analogous theorems.
Reviewer: M.Fuchs

##### MSC:
 58E20 Harmonic maps, etc. 49Q99 Manifolds and measure-geometric topics
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