Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions

Giuseppe Geymonat

doi:10.5802/ambp.232

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, Tome 14 (2007) no. 2, pp. 187-197.

Résumé

A famous theorem of E. Gagliardo gives the characterization of traces for Sobolev spaces $W^{1, p} (Ω)$ for $1 \leq p < \infty$ when $Ω \subset ℝ^{N}$ is a Lipschitz domain. The extension of this result to $W^{m, p} (Ω)$ for $m \geq 2$ and $1 < p < \infty$ 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 $m \geq 3$ and we prove how the local compatibility conditions can be derived.

MR Zbl

DOI : 10.5802/ambp.232

Affiliations des auteurs :

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

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     title = {Trace {Theorems} for {Sobolev} {Spaces} on {Lipschitz} {Domains.} {Necessary} {Conditions}},
     journal = {Annales math\'ematiques Blaise Pascal},
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Giuseppe Geymonat. Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions. Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 2, pp. 187-197. doi : 10.5802/ambp.232. https://ambp.centre-mersenne.org/articles/10.5802/ambp.232/

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