By A. Harnack
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Additional info for An Intro to the Study of the Elements of the Diff and Int Calculus
However, it is also possible to assign to each f ∈ W k,p (Ω, RN ) boundary values in the space Lp (∂Ω) (and even better), and this is a bounded, linear operation, which acts as the restriction operator to ∂Ω whenever f is in addition continuous on Ω. 7]. (iii) Approximation by smooth functions: The Sobolev spaces can also be deﬁned via approximation by smooth functions. One way, as mentioned before, is the deﬁnition as the closure of C ∞ (Ω, RN ) ∩ W k,p (Ω, RN ) in W k,p (Ω, RN ). e. C ∞ (Ω, RN ) ∩ W k,p (Ω, RN ) is dense in W k,p (Ω, RN ).
58]. Restricting ourselves only to embeddings into Lebesgue and H¨older spaces and to fractional Sobolev spaces with order of diﬀerentiability s ≤ 1, these results amount to the following statement. 67 Let Ω be a bounded domain in Rn with Lipschitz boundary. Furthermore, let s ∈ (0, 1], p ∈ (1, ∞) and assume f ∈ W s,p (Ω, RN ). Then the following statements are true: (i) If n > sp, then f ∈ Lt (Ω, RN ) for all t ≤ np/(n − sp). (ii) If n = sp, then f ∈ Lt (Ω, RN ) for all t < ∞. (iii) If n < sp, then f ∈ C(Ω, RN ).
I) For every function f ∈ W01,p (Ω, RN ) we have f Lp (Ω,RN ) ≤ c(n, N, p, Ω) Df Lp (Ω,RN n ) . 40 1 Preliminaries (ii) If Ω is connected with Lipschitz-boundary, then for every function f ∈ W 1,p (Ω, RN ) we have f − (f )Ω Lp (Ω,RN ) ≤ c(n, N, p, Ω) Df Lp (Ω,RN n ) . 51 (i), applied with p if p < n and with n+p ∈ [1, n) otherwise. In order to derive the inequality in (ii), we may assume without loss of generality (f )Ω = 0 since the claimed inequality is invariant under the addition of constants to f .