Bethuel, F. A characterization of maps in \(H^ 1(B^ 3,S^ 2)\) which can be approximated by smooth maps. (English) Zbl 0708.58004 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7, No. 4, 269-286 (1990). 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 Cited in 1 ReviewCited in 41 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML 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, {\it The approximation problem for Sobolev maps between two manifolds}, to appear. · Zbl 0756.46017 [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, {\it Minimizing p−harmonic maps into spheres}, preprint. · Zbl 0677.58021 [6] F. Helein, {\it 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.