Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

2024/12/17 by Detaille, Antoine, Van Schaftingen, Jean · 2 citations
#46T10 #58C25 (Secondary) #58D15 (Primary) 46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2412.12889

Abstract

For any integer p ≥ 2 , we construct a compact Riemannian manifold N such that if dim M > p , there is a map in the Sobolev space of mappings W^1,p (M, N) which is not a weak limit of smooth maps into N due to a mechanism of analytical obstruction. For p = 4n - 1 , the target manifold can be taken to be the sphere \mathbbS²ⁿ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for p = 4n -1 = 3. The results extend to higher order Sobolev spaces W^s,p , with s ∈ ℝ , s ≥ 1 , sp ∈ ℕ, and sp ≥ 2 .

Cited by

• Equi-integrable approximation of Sobolev mappings between manifolds
• Heterotopic energy for Sobolev mappings

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