Trace Theorems for Sobolev Spaces on Lipschitz Domains. Necessary Conditions

Abstract

cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques Abstract 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.