Abstract
Brezis and Mironescu have announced several years ago that for a compact manifold \({N^n \subset \mathbb{R}^\upsilon}\) and for real numbers 0 < s < 1 and \({1 \leq p < \infty}\), the class \({C^\infty(\overline{Q}^m;N^n)}\) of smooth maps on the cube with values into N n is dense with respect to the strong topology in the Sobolev space \({W^{s,p}(Q^m;N^n)}\) when the homotopy group \({\pi_{{\lfloor}sp{\rfloor}}(N^n)}\) of order \({\lfloor sp \rfloor}\) is trivial. The proof of this beautiful result is long and rather involved. Under the additional assumption that N n is \({\lfloor sp \rfloor}\) simply connected, we give a shorter and different proof of their result. Our proof for \({sp \geq 1}\) is based on the existence of a retraction of \({\mathbb{R}^\upsilon}\) onto Nn except for a small subset in the complement of N n and on the Gagliardo–Nirenberg interpolation inequality for maps in \({W^{1,q} \cap L^\infty}\). In contrast, the case \({sp < 1}\) relies on the density of step functions on cubes in W s,p.
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A l’infatigable Haïm Brezis pour ses 70 ans, avec admiration
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Bousquet, P., Ponce, A.C. & Van Schaftingen, J. Strong approximation of fractional Sobolev maps. J. Fixed Point Theory Appl. 15, 133–153 (2014). https://doi.org/10.1007/s11784-014-0172-5
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DOI: https://doi.org/10.1007/s11784-014-0172-5