2013/10/22 by Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen
Mathematics · #Advanced Harmonic Analysis Research #Class (philosophy) #Complement (music) #Cube (algebra) #Geometric Analysis and Curvature Flows #Interpolation (computer graphics) #Manifold (fluid mechanics) #Nonlinear Partial Differential Equations #Order (exchange) #Sobolev inequality #Sobolev space #math.FA #msc:46E35 #msc:46T20 #msc:58D15
paper · pdf · doi:10.1007/s11784-014-0172-5
published as J. Fixed Point Theory Appl. 15 (2014), no. 1, 133-153
arxiv created 2013/10/22 · openalex publication_date 2014/03/01 · arxiv updated 2015/01/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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\lfloor sp \rfloor(Nn) of order \lfloor sp \rfloor is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that Nn is \lfloor sp \rfloor 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.