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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
##### Keywords:
density of smooth maps; Sobolev space
Full Text:
##### References:
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