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A characterization of maps in \(H^ 1(B^ 3,S^ 2)\) which can be approximated by smooth maps. (English) Zbl 0708.58004
Consider the Sobolev space \(H^ 1(B^ 3,S^ 2)=\{u\in H^ 1(B^ 3,R^ 3):\) \(u(x)\in S^ 2\) a.e.\(\}\) where \(B^ 3\) is the open unit ball and \(S^ 2\) the unit sphere in \({\mathbb{R}}^ 3\). Given a map \(u\in H^ 1(B^ 3,S^ 2)\), denote \(D(u)=(u\cdot u_ y\bigwedge u_ z\), \(u\cdot u_ z\bigwedge u_ x\), \(u\cdot u_ x\bigwedge u_ y)\). For u regular except at most at a finite number of point singularities \(a_ 1,...,a_ n\in B^ 3\), it is div D(u)\(=4\pi \sum^{n}_{i=1}\deg (u,a_ i)\delta_{a_ i}\) where \(\deg (u,a_ i)\) denotes the Brower degree of u at \(a_ i.\)
The following characterization is the main result of the paper:
Theorem. A map \(u\in H^ 1(B^ 3,S^ 2)\) can be approximated by smooth maps if and only if div D(u)\(=0.\)
Moreover, it is shown that \(C^{\infty}(B^ 3,S^ 2)\) is dense (and in fact, sequentially dense) for the weak topology in \(H^ 1(B^ 3,S^ 2)\). Some partial results for \({\mathbb{R}}^ n\) are mentioned, too.
Reviewer: J.Durdil

MSC:
58C25 Differentiable maps on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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References:
[1] H. Brezis, private communication.
[2] Brezis, H.; Coron, J. M.; Lieb, E. H., Harmonic maps with defects, Comm. Math. Phys., t. 107, 649-705, (1986) · Zbl 0608.58016
[3] F. Bethuel, The approximation problem for Sobolev maps between two manifolds, to appear.
[4] Bethuel, F.; Zheng, X., Density of smooth functions between two manifolds in Sobolev spaces, J. Func. Anal., t. 80, 60-75, (1988) · Zbl 0657.46027
[5] J. M. Coron and R. Gulliver, Minimizing p−harmonic maps into spheres, preprint. · Zbl 0677.58021
[6] F. Helein, Approximations of Sobolev maps between an open set and an euclidean sphere, boundary data, and singularities, preprint. · Zbl 0659.35002
[7] Schoen, R.; Uhlenbeck, K., A regularity theory for harmonic maps, J. Diff. Geom., t. 17, 307-335, (1982) · Zbl 0521.58021
[8] White, B., Infima of energy functionals in homotopy classes, J. Diff. Geom., t. 23, 127-142, (1986) · Zbl 0588.58017
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