JournalsjemsVol. 17, No. 4pp. 763–817

Strong density for higher order Sobolev spaces into compact manifolds

  • Pierre Bousquet

    Université de Toulouse, France
  • Augusto C. Ponce

    Université Catholique de Louvain, Belgium
  • Jean Van Schaftingen

    Université Catholique de Louvain, Belgium
Strong density for higher order Sobolev spaces into compact manifolds cover
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Abstract

Given a compact manifold NnN^n, an integer kNk \in \mathbb{N}_* and an exponent 1p<1 \le p < \infty, we prove that the class C(Qm;Nn)C^\infty(\overline{Q}^m; N^n) of smooth maps on the cube with values into NnN^n is dense with respect to the strong topology in the Sobolev space Wk,p(Qm;Nn)W^{k, p}(Q^m; N^n) when the homotopy group πkp(Nn)\pi_{\lfloor kp \rfloor}(N^n) of order kp\lfloor kp \rfloor is trivial. We also prove density of maps that are smooth except for a set of dimension mkp1m - \lfloor kp \rfloor - 1, without any restriction on the homotopy group of NnN^n.

Cite this article

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen, Strong density for higher order Sobolev spaces into compact manifolds. J. Eur. Math. Soc. 17 (2015), no. 4, pp. 763–817

DOI 10.4171/JEMS/518