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By Kenneth S. Miller.

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Extra resources for An Introduction to the Calculus of Finite Differences and Difference Equations

Example text

Then we have f ∈ W θ ,p (Ω ) for every θ ∈ (0, θ) and Ω Ω . Moreover, there holds |f (x) − f (y)| Ω Ω |x − y| n+θ p p dx dy ≤ c(n, p) K p d(θ−θ )p |Ω | + n+θ p θ−θ d for d := min{1, dist(Ω , ∂Ω), dist(Ω , ∂Ω )}. 2 Criteria for Weak Differentiability 31 Proof We may suppose that f is smooth, since the general statement then n n follows via approximation. For a vector v = we write s=1 vs es ∈ R k (k) (0) v = s=1 vs es for k = 1, . . , n and v = 0. Then we can decompose the difference f (x + v) − f (x) into differences of f along the coordinate directions and find n n τs,vs f (x + v (s−1) ) ≤ |f (x + v) − f (x)| = s=1 |τs,vs f (x + v (s−1) )| s=1 whenever x + v (s−1) ∈ Ω for all s ∈ {0, 1, .

17 (to the remaining n − 1 factors with all exponents equal to n − 1) and finally Fubini’s theorem. This gives n R (2|f (x)|) n−1 dx1 ≤ R |D1 f (ξ1 , x2 , . . , xn |) dξ1 1 n−1 n × R i=2 R |Di f (x1 , . . , ξi , . . , xn )| dξi 1 n−1 dx1 34 1 ≤ R |D1 f (x1 , x2 , . . , xn |) dx1 Preliminaries 1 n−1 n × R2 i=2 |Di f (x1 , . . , ξi , . . 14) is true for = 1. 14) to be true for all i ≤ − 1 and some ∈ {2, . . , n − 1}. 14) −1 with respect to x . 14) is independent of x . 14) for all ∈ {1, . .

G. 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 ).

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