We consider the problem of strong density of smooth maps in the Sobolev space $ W^{ s,p}(Q^{ m};\mathcal{ N}) $, where $ 0 < s < +\infty $, $ 1 \leq p < +\infty $, $ Q^{ m} $ is the unit cube in $ \mathbb{ R}^{ m} $, and $ \mathcal{ N} $ is a smooth compact connected Riemannian manifold without boundary. Our main result fully answers the strong density problem in the whole range $ 0 < s < +\infty $: the space $ \mathcal{ C}^{ \infty}(\overline{ Q}^{ m};\mathcal{ N}) $ is dense in $ W^{ s,p}(Q^{ m};\mathcal{ N}) $ if and only if $ \pi_{ [sp]}(\mathcal{ N}) = \{ 0\} $. This completes the results of Bethuel ($ s=1 $), Brezis and Mironescu ($ 0 < s < 1 $), and Bousquet, Ponce, and Van Schaftingen ($ s = 2 $, $ 3 $, ...). We also consider the case of more general domains $ \Omega $, in the setting studied by Hang and Lin when $ s = 1 $.
Categories: math.FA