Composition in Fractional Sobolev Spaces

نویسندگان

  • HAIM BREZIS
  • PETRU MIRONESCU
چکیده

1. Introduction. A classical result about composition in Sobolev spaces asserts that if u ∈ W k,p (Ω)∩L ∞ (Ω) and Φ ∈ C k (R), then Φ • u ∈ W k,p (Ω). Here Ω denotes a smooth bounded domain in R N , k ≥ 1 is an integer and 1 ≤ p < ∞. This result was first proved in [13] with the help of the Gagliardo-Nirenberg inequality [14]. In particular if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R) then Φ • u ∈ W k,p since W k,p ⊂ L ∞ by the Sobolev embedding theorem. When kp = N the situation is more delicate since W k,p is not contained in L ∞. However the following result still holds (see [2],[3])

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تاریخ انتشار 2004