Strong approximation of fractional Sobolev maps

Original · EN

Brezis and Mironescu have announced several years ago that for a compact manifold Nⁿ ⊂ Rν and for real numbers 0 < s < 1 and 1 ≤ p < ∞ the class C∞(Qᵐ; Nⁿ) of smooth maps on the cube with values into Nⁿ is dense with respect to the strong topology in the Sobolev space Wˢ, ᵖ(Qᵐ; Nⁿ) when the homotopy group π sp (Nⁿ) of order sp is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that Nⁿ is sp simply connected, we give a shorter proof of their result. Our proof for sp ≥ 1 is based on the existence of a retraction of Rν onto Nⁿ except for a small subset in the complement of Nⁿ and on the Gagliardo-Nirenberg interpolation inequality for maps in W¹, q ∩ L∞. In contrast, the case sp < 1 relies on the density of step functions on cubes in Wˢ, ᵖ.