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


58C25 Differentiable maps on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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