Norm of Sobolev space Wsp(Ω)

  • When s0

    • When sN0, write it as k

      uWkp(Ω)={(|α|kDαupLp(Ω))1pp[1,)max|α|kDαuL(Ω)p=
    • When s>0 and is fractional, let it be s=k+κ, with k=s and κ(0,1) uWsp(Ω)=(upWkp(Ω)+|u|pWsp(Ω))1p

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

      |u|pWsp(Ω)=|α|=kΩΩ|Dαu(x)Dαu(y)|p|xy|d+pκdxdy
  • When s<0 and p(1,)

    • Wsp(Ω)=(Wsq) with 1p+1q=1. Then its norm is the operator norm uWsp(Ω)=sup0vWsq(Ω)|u,vΩ|vWsq(Ω)
    • Wsp(Ω)=(Wsq(Ω)), its operator norm is uWsp(Ω)=sup0vWsq(Ω)|u,vΩ|vWsq(Ω)

Generate Sobolev spaces by taking closure

  • Wsp(Ω)=¯C(Ω)Wsp(Ω) with s0 and p[1,]
  • Wsp(Ω)=¯C0(Ω)Wsp(Ω) with s0 and p[1,]
  • ˜Hs(Ω)=¯C0(Ω)Hs(Rd)=¯C0(Ω)Ws2(Rd) with sR
  • Hs0(Ω)=¯C0(Ω)Hs(Ω)

Generate Sobolev spaces by taking restriction

  • Hs(Ω)={v=˜v|Ω:˜vHs(Rd)}

Duality relation between Sobolev spaces

  • Wsp(Ω)=(Wsq) with s<0 and p(1,)
  • Wsp(Ω)=(Wsq(Ω)) with s<0 and p(1,)
  • ˜Hs(Ω)=[Hs(Ω)] for all sR when Ω is a Lipschitz domain
  • Hs(Ω)=[˜Hs(Ω)] for all sR when Ω is a Lipschitz domain
  • Hs(Γ)=[Hs(Γ)] for s<0

Equivalent Sobolev spaces

  • Hs(Rd)=Ws2(Rd) for all sR
  • Hs(Ω)=Ws2(Ω) for all s>0 when Ω is a Lipschitz domain
  • ˜Hs(Ω)=Hs0(Ω) for s0 and s{12,32,52,}, when Ω is a Lipschitz domain