Norm of Sobolev space Wsp(Ω)

  • When s≥0

    • When s∈N0, write it as k

      ‖u‖Wkp(Ω)={(∑|α|≤k‖Dαu‖pLp(Ω))1pp∈[1,∞)max|α|≤k‖Dαu‖L∞(Ω)p=∞

    • When s>0 and is fractional, let it be s=k+κ, with k=⌊s⌋ and κ∈(0,1) ‖u‖Wsp(Ω)=(‖u‖pWkp(Ω)+|u|pWsp(Ω))1p

      where |u|Wsp(Ω) is the Sobolev-Slobodeckii semi-norm

      |u|pWsp(Ω)=∑|α|=k∫Ω∫Ω|Dαu(x)−Dαu(y)|p|x−y|d+pκdxdy

  • When s<0 and p∈(1,∞)

    • Wsp(Ω)=(∘W−sq)′ with 1p+1q=1. Then its norm is the operator norm ‖u‖Wsp(Ω)=sup0≠v∈∘W−sq(Ω)|⟨u,v⟩Ω|‖v‖∘W−sq(Ω)
    • ∘Wsp(Ω)=(W−sq(Ω))′, its operator norm is ‖u‖∘Wsp(Ω)=sup0≠v∈W−sq(Ω)|⟨u,v⟩Ω|‖v‖W−sq(Ω)

Generate Sobolev spaces by taking closure

  • Wsp(Ω)=¯C∞(Ω)‖⋅‖Wsp(Ω) with s≥0 and p∈[1,∞]
  • ∘Wsp(Ω)=¯C∞0(Ω)‖⋅‖Wsp(Ω) with s≥0 and p∈[1,∞]
  • ˜Hs(Ω)=¯C∞0(Ω)‖⋅‖Hs(Rd)=¯C∞0(Ω)‖⋅‖Ws2(Rd) with s∈R
  • Hs0(Ω)=¯C∞0(Ω)‖⋅‖Hs(Ω)

Generate Sobolev spaces by taking restriction

  • Hs(Ω)={v=˜v|Ω:˜v∈Hs(Rd)}

Duality relation between Sobolev spaces

  • Wsp(Ω)=(∘W−sq)′ with s<0 and p∈(1,∞)
  • ∘Wsp(Ω)=(W−sq(Ω))′ with s<0 and p∈(1,∞)
  • ˜Hs(Ω)=[H−s(Ω)]′ for all s∈R when Ω is a Lipschitz domain
  • Hs(Ω)=[˜H−s(Ω)]′ for all s∈R when Ω is a Lipschitz domain
  • Hs(Γ)=[H−s(Γ)]′ for s<0

Equivalent Sobolev spaces

  • Hs(Rd)=Ws2(Rd) for all s∈R
  • Hs(Ω)=Ws2(Ω) for all s>0 when Ω is a Lipschitz domain
  • ˜Hs(Ω)=Hs0(Ω) for s≥0 and s≠{12,32,52,⋯}, when Ω is a Lipschitz domain