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Closure of smooth maps in \(W^{1,p}(B^3;S^2)\). (English) Zbl 1240.46063
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 \(2<p<3\).

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
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