Ponce, Augusto C.; Van Schaftingen, Jean Closure of smooth maps in \(W^{1,p}(B^3;S^2)\). (English) Zbl 1240.46063 Differ. Integral Equ. 22, No. 9-10, 881-900 (2009). The paper deals with an alternative approach to the rather surprising phenomenon of density of smooth (up to the boundary of \(B^3\)) functions in \(W^{1,p}(B^3;S^2)\) in dependence on the exponent \(p\). While the density is true if \(p\geq 3\) or \(1\leq p<2\), it generally fails in the case \(2\leq p<3\). In the last case, the necessary and sufficient condition for the approximation of \(u\in W^{1,p}(B^3;S^2)\) by a smooth function reads as the vanishing of the distributional Jacobian of \(u\). The authors suitably use and extend procedures known for the case \(p=2\) for identification of the singularities responsible for spoiling the approximation properties to get the claim in the whole range \(2<p<3\). Reviewer: Miroslav Krbec (Praha) Cited in 1 ReviewCited in 7 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46T30 Distributions and generalized functions on nonlinear spaces 58D15 Manifolds of mappings Keywords:Sobolev spaces on manifolds; density of smooth functions PDFBibTeX XMLCite \textit{A. C. Ponce} and \textit{J. Van Schaftingen}, Differ. Integral Equ. 22, No. 9--10, 881--900 (2009; Zbl 1240.46063) Full Text: arXiv