Analytical obstructions to weak approximation by smooth maps exist for every integer Sobolev exponent p≥2, and for spheres S^{2n} when p=4n-1.
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Analytical obstructions to the weak approximation of Sobolev mappings into manifolds
Analytical obstructions to weak approximation by smooth maps exist for every integer Sobolev exponent p≥2, and for spheres S^{2n} when p=4n-1.