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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 $$2<p<3$$.

##### 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
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