Strong density for higher order Sobolev spaces into compact manifolds
(2015) Journal of the European Mathematical Society — Vol. 17, n° 4, p. 763-817 (2015)
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- Abstract
- Given a compact manifold \(N^n\), an integer \(k \in \mathbb{N}_*\) and an exponent \(1 \le p < \infty\), we prove that 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 \(W^{k, p}(Q^m; N^n)\) when the homotopy group \(\pi_{\lfloor kp \rfloor}(N^n)\) of order \(\lfloor kp \rfloor\) is trivial. We also prove the density of maps that are smooth except for a set of dimension \(m - \lfloor kp \rfloor - 1\), without any restriction on the homotopy group of \(N^n\).
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Bousquet, P., Ponce, A., & Van Schaftingen, J. (2015). Strong density for higher order Sobolev spaces into compact manifolds. Journal of the European Mathematical Society, 17(4), 763-817. https://doi.org/10.4171/JEMS/518 (Original work published 2015)