Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions

Giuseppe Geymonat[1]

  • [1] Laboratoire de Mécanique et de Génie Civil, UMR 5508 CNRS, Université Montpellier II Place Eugène Bataillon 34695 Montpellier Cedex 5 France

Annales mathématiques Blaise Pascal (2007)

  • Volume: 14, Issue: 2, page 187-197
  • ISSN: 1259-1734

Abstract

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A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces W 1 , p Ω for 1 ≤ p < ∞ when Ω ⊂ ℝ N is a Lipschitz domain. The extension of this result to W m , p Ω for m ≥ 2 and 1 < p < ∞ is now well-known when Ω is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compatibility conditions at the vertices, edges, .... Some recent papers give the characterization for general Lipschitz domains for m=2 in terms of global compatibility conditions. Here we give the necessary compatibility conditions for m ≥ 3 and we prove how the local compatibility conditions can be derived.

How to cite

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Geymonat, Giuseppe. "Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions." Annales mathématiques Blaise Pascal 14.2 (2007): 187-197. <http://eudml.org/doc/10545>.

@article{Geymonat2007,
abstract = {A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces $W^\{1,\, p\}\left(\Omega \right)$ for $1\le p&lt;\infty $ when $\Omega \subset \mathbb\{R\}^\{N\}$ is a Lipschitz domain. The extension of this result to $W^\{m,\, p\}\left(\Omega \right)$ for $m\ge 2$ and $1&lt;p&lt;\infty $ is now well-known when $\Omega $ is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compatibility conditions at the vertices, edges, .... Some recent papers give the characterization for general Lipschitz domains for m=2 in terms of global compatibility conditions. Here we give the necessary compatibility conditions for $m\ge 3$ and we prove how the local compatibility conditions can be derived.},
affiliation = {Laboratoire de Mécanique et de Génie Civil, UMR 5508 CNRS, Université Montpellier II Place Eugène Bataillon 34695 Montpellier Cedex 5 France},
author = {Geymonat, Giuseppe},
journal = {Annales mathématiques Blaise Pascal},
language = {eng},
month = {7},
number = {2},
pages = {187-197},
publisher = {Annales mathématiques Blaise Pascal},
title = {Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions},
url = {http://eudml.org/doc/10545},
volume = {14},
year = {2007},
}

TY - JOUR
AU - Geymonat, Giuseppe
TI - Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions
JO - Annales mathématiques Blaise Pascal
DA - 2007/7//
PB - Annales mathématiques Blaise Pascal
VL - 14
IS - 2
SP - 187
EP - 197
AB - A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces $W^{1,\, p}\left(\Omega \right)$ for $1\le p&lt;\infty $ when $\Omega \subset \mathbb{R}^{N}$ is a Lipschitz domain. The extension of this result to $W^{m,\, p}\left(\Omega \right)$ for $m\ge 2$ and $1&lt;p&lt;\infty $ is now well-known when $\Omega $ is a smooth domain. The situation is more complicated for polygonal and polyhedral domains since the characterization is given only in terms of local compatibility conditions at the vertices, edges, .... Some recent papers give the characterization for general Lipschitz domains for m=2 in terms of global compatibility conditions. Here we give the necessary compatibility conditions for $m\ge 3$ and we prove how the local compatibility conditions can be derived.
LA - eng
UR - http://eudml.org/doc/10545
ER -

References

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  1. R. A. Adams, J. J. F. Fournier, Sobolev spaces. Second edition, (2003), Academic Press, New York Zbl1098.46001
  2. A. Buffa, G. Geymonat, On the traces of functions in W 1 , p ( Ω ) for Lipschitz domains in ℝ 3 , C. R. Acad. Sci. Paris, Série I 332 (2001), 699-704 Zbl0987.46036MR1843191
  3. A. Buffa, Jr P. Ciarlet, On traces for functional spaces related to Maxwell’s equations. Part I: an integration by parts formula in Lipschitz Polyedra, Math. Meth. Appl. Sci. 24 (2001), 9-30 Zbl0998.46012
  4. Zhonghai Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proc. A. M. S. 124 (1996), 591-600 Zbl0841.46021MR1301021
  5. R. G. Durán, M. A. Muschietti, On the traces of W 2 , p Ω for a Lipschitz domain, Rev. Mat. Complutense XIV (2001), 371-377 Zbl1029.46031MR1871302
  6. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n-variabili, Rend. Sem. Mat. Univ. Padova 27 (1957), 284-305 Zbl0087.10902MR102739
  7. G. Geymonat, F. Krasucki, On the existence of the airy function in Lipschitz domains. Application to the traces of H 2 , C. R. Acad. Sci. Paris, Série I 330 (2000), 355-360 Zbl0945.35065MR1751670
  8. P. Grisvard, Elliptic boundary value problems in nonsmooth domains, (1985), Pitman, London Zbl0695.35060
  9. J. Nečas, Les méthodes directes en théorie des équations elliptiques, (1967), Masson, Paris MR227584

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