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These functions have a well-defined restriction or trace on Ωl. The imbedding result means that the functions un|Ωl converge to a function in Wm, p(Ωl). SOBOLEV SPACES, VECTOR FUNCTION SPACES AND REGULARITY 41 The imbedding of Wm + j, p(Ω) in a space of continuous functions is understood in the sense that there is a member of the equivalence class of functions u ∈ Wm + j, p(Ω) with the required continuity. The following statement of the famous Sobolev imbedding theorem is from [2]. 5Let Ω ⊂ RN be a bounded domain with Lipschitz continuous boundary and suppose m, j are non-negative integers and 1 ≤ l ≤ N.

27 of [215]. 2 holds for fractionalorder spaces (cf. [151] for a discussion of fractional-order Sobolev spaces on Lipschitz domains). We should point out that the results quoted for fractional-order spaces are a very small selection of the known results (cf. [2]). We shall denote by H-1(Ω) the dual space of with the usual dual norm. 1 Trace spaces We have one further topic in basic Sobolev space theory to discuss, in particular, the way in which boundary values or traces of functions are handled.

6) when m = 1, p = 2, j = 0, l = N and Ω0 = Ω states that H1(Ω) is compactly imbedded in L2(Ω) for N = 2, 3. This observation underlies the analysis of the Helmholtz equation. Unfortunately, in the analysis of boundary values of functions, and in discussing the regularity of solutions of Maxwell's equations, it is necessary to use Sobolev spaces of fractional order. Following [237], we define the spaces Ws, p(Ω), 1 ≤ p < ∞, s ∈ R and s ≥ 0 as follows. Let m ∈ Z+ and suppose s = m + σ, 42 SOBOLEV SPACES, VECTOR FUNCTION SPACES AND REGULARITY where σ ∈ R and 0 < σ < 1.

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