Strong density for higher order Sobolev spaces into compact manifolds

Details Strong density for higher order Sobolev spaces into compact manifolds Strong density for higher order Sobolev spaces into compact manifolds

2012-03-16

arxiv.org/abs/1203.3721v1

Mathematics

Functional Analysis

48 pages

Scientific paper

Given a compact manifold (N^n), (k \in \mathbb{N}_*) and (1 \le p < \infty), we prove that the class (C^\infty(\bar{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)) if 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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