Retractions onto the Space of Continuous Divergence-free Vector Fields

Philippe Bouafia[1]

  • [1] Université Paris Sud 11, Département de Mathématiques, 91405 Orsay Cedex

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 4, page 767-779
  • ISSN: 0240-2963

Abstract

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We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of m -charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset X ⊆ ℝ n satisfying a mild geometric condition, there is no uniformly continuous representation operator for m -charges in X .

How to cite

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Bouafia, Philippe. "Retractions onto the Space of Continuous Divergence-free Vector Fields." Annales de la faculté des sciences de Toulouse Mathématiques 20.4 (2011): 767-779. <http://eudml.org/doc/219822>.

@article{Bouafia2011,
abstract = {We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of $m$-charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset $X \subseteq \mathbb\{R\}^n$ satisfying a mild geometric condition, there is no uniformly continuous representation operator for $m$-charges in $X$.},
affiliation = {Université Paris Sud 11, Département de Mathématiques, 91405 Orsay Cedex},
author = {Bouafia, Philippe},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {vector fields; -charges},
language = {eng},
month = {7},
number = {4},
pages = {767-779},
publisher = {Université Paul Sabatier, Toulouse},
title = {Retractions onto the Space of Continuous Divergence-free Vector Fields},
url = {http://eudml.org/doc/219822},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Bouafia, Philippe
TI - Retractions onto the Space of Continuous Divergence-free Vector Fields
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/7//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 4
SP - 767
EP - 779
AB - We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of $m$-charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset $X \subseteq \mathbb{R}^n$ satisfying a mild geometric condition, there is no uniformly continuous representation operator for $m$-charges in $X$.
LA - eng
KW - vector fields; -charges
UR - http://eudml.org/doc/219822
ER -

References

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  6. Federer (H.).— Geometric Measure Theory. Springer-Verlag, New York (1971). Zbl0176.00801MR257325
  7. Godefroy (G.) and Kalton (N.J.).— Lipschitz-free banach spaces. Studia Mathematica, 159 (1), p. 121141 (2003). Zbl1059.46058MR2030906
  8. De Pauw (T.), Moonens (L.) and Pfeffer (W.F.).— Charges in middle dimensions. J. Math. Pures Appl., 92, p. 86112 (2009). Zbl1166.49039MR2541148
  9. De Pauw (T.) and Torres (M.).— On the distributional divergence of vector fields vanishing at infinity. Proc. Roy. Soc. Edinburgh, 141A, p. 6576 (2011). Zbl1213.35181MR2773439
  10. Wojtaszczyk (P.).— Banach Spaces for Analysts. Cambridge University Press (1996). Zbl0724.46012MR1144277

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