Density of smooth maps for fractional Sobolev spaces W s , p into ℓ simply connected manifolds when s ≥ 1

Pierre Bousquet[1]; Augusto C. Ponce[2]; Jean Van Schaftingen[2]

  • [1] Aix-Marseille Université, Laboratoire d’analyse, topologie, probabilités UMR7353, CMI 39, Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France
  • [2] Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium

Confluentes Mathematici (2013)

  • Volume: 5, Issue: 2, page 3-22
  • ISSN: 1793-7434

Abstract

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Given a compact manifold N n ⊂ ℝ ν and real numbers s ≥ 1 and 1 ≤ p < ∞ , we prove that the class C ∞ ( Q ¯ m ; N n ) of smooth maps on the cube with values into N n is strongly dense in the fractional Sobolev space W s , p ( Q m ; N n ) when N n is ⌊ s p ⌋ simply connected. For s p integer, we prove weak sequential density of C ∞ ( Q ¯ m ; N n ) when N n is s p - 1 simply connected. The proofs are based on the existence of a retraction of ℝ ν onto N n except for a small subset of N n and on a pointwise estimate of fractional derivatives of composition of maps in W s , p ∩ W 1 , s p .

How to cite

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Bousquet, Pierre, Ponce, Augusto C., and Van Schaftingen, Jean. "Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$." Confluentes Mathematici 5.2 (2013): 3-22. <http://eudml.org/doc/275426>.

@article{Bousquet2013,
abstract = {Given a compact manifold $N^n \subset \{\mathbb\{R\}\}^\nu $ and real numbers $s \ge 1$ and $1 \le p &lt; \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 strongly dense in the fractional Sobolev space $W^\{s, p\}(Q^m; N^n)$ when $N^n$ is $\lfloor sp\rfloor $ simply connected. For $sp$ integer, we prove weak sequential density of $C^\infty (\overline\{Q\}^m; N^n)$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\{\mathbb\{R\}\}^\nu $ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^\{s, p\} \cap W^\{1, sp\}$.},
affiliation = {Aix-Marseille Université, Laboratoire d’analyse, topologie, probabilités UMR7353, CMI 39, Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France; Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium; Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgium},
author = {Bousquet, Pierre, Ponce, Augusto C., Van Schaftingen, Jean},
journal = {Confluentes Mathematici},
keywords = {Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness; strong density},
language = {eng},
number = {2},
pages = {3-22},
publisher = {Institut Camille Jordan},
title = {Density of smooth maps for fractional Sobolev spaces $W^\{s, p\}$ into $\ell $ simply connected manifolds when $s \ge 1$},
url = {http://eudml.org/doc/275426},
volume = {5},
year = {2013},
}

TY - JOUR
AU - Bousquet, Pierre
AU - Ponce, Augusto C.
AU - Van Schaftingen, Jean
TI - Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell $ simply connected manifolds when $s \ge 1$
JO - Confluentes Mathematici
PY - 2013
PB - Institut Camille Jordan
VL - 5
IS - 2
SP - 3
EP - 22
AB - Given a compact manifold $N^n \subset {\mathbb{R}}^\nu $ and real numbers $s \ge 1$ and $1 \le p &lt; \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 strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp\rfloor $ simply connected. For $sp$ integer, we prove weak sequential density of $C^\infty (\overline{Q}^m; N^n)$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of ${\mathbb{R}}^\nu $ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.
LA - eng
KW - Strong density; weak sequential density; Sobolev maps; fractional Sobolev spaces; simply connectedness; strong density
UR - http://eudml.org/doc/275426
ER -

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