Norm of Sobolev space \(W_p^s(\Omega)\)

  • When \(s\geq 0\)

    • When \(s\in\mathbb{N}_0\), write it as \(k\)

      \[\norm{u}_{W_p^k(\Omega)} = \begin{cases} \displaystyle{\left( \sum_{\abs{\alpha}\leq k} \norm{D^{\alpha}u}_{L_p(\Omega)}^p \right)^{\frac{1}{p}}} & p \in [1,\infty) \\ \displaystyle{\max_{\abs{\alpha}\leq k} \norm{D^{\alpha}u}_{L_{\infty}(\Omega)}} & p = \infty \end{cases}\]

    • When \(s > 0\) and is fractional, let it be \(s=k+\kappa\), with \(k=\lfloor s \rfloor\) and \(\kappa\in(0,1)\) \(\norm{u}_{W_p^s(\Omega)} = \left( \norm{u}_{W_p^k(\Omega)}^p + \abs{u}_{W_p^s(\Omega)}^p \right)^{\frac{1}{p}}\)

      where \(\abs{u}_{W_p^s(\Omega)}\) is the Sobolev-Slobodeckii semi-norm

      \[\abs{u}_{W_p^s(\Omega)}^p = \sum_{\abs{\alpha}=k}\int_{\Omega}\int_{\Omega} \frac{\abs{D^{\alpha}u(x) - D^{\alpha}u(y)}^p}{\abs{x-y}^{d+p\kappa}} \intd x \intd y\]

  • When \(s < 0\) and \(p \in (1, \infty)\)

    • \(W_p^s(\Omega) = \left( \overset{\circ}{W}_q^{-s} \right)'\) with \(\frac{1}{p} + \frac{1}{q}=1\). Then its norm is the operator norm \(\norm{u}_{W_p^s(\Omega)}=\sup_{0\neq v \in \overset{\circ}{W}_q^{-s}(\Omega)} \frac{\abs{\left\langle u,v \right\rangle_{\Omega}}}{\norm{v}_{\overset{\circ}{W}_q^{-s}(\Omega)}}\)
    • \(\overset{\circ}{W}_p^s(\Omega) = \left( W_q^{-s}(\Omega) \right)'\), its operator norm is \(\norm{u}_{\overset{\circ}{W}_p^s(\Omega)} = \sup_{0\neq v \in W_{q}^{-s}(\Omega)} \frac{\abs{\left\langle u,v \right\rangle_{\Omega}}}{\norm{v}_{W_q^{-s}(\Omega)}}\)

Generate Sobolev spaces by taking closure

  • \(W_p^s(\Omega) = \overline{C^{\infty}(\Omega)}^{\norm{\cdot}_{W_p^s(\Omega)}}\) with \(s \geq 0\) and \(p \in [1, \infty]\)
  • \(\overset{\circ}{W}_p^s(\Omega) = \overline{C_0^{\infty}(\Omega)}^{\norm{\cdot}_{W_p^s(\Omega)}}\) with \(s \geq 0\) and \(p \in [1, \infty]\)
  • \(\widetilde{H}^s(\Omega) = \overline{C_0^{\infty}(\Omega)}^{\norm{\cdot}_{H^s(\mathbb{R}^d)}} = \overline{C_0^{\infty}(\Omega)}^{\norm{\cdot}_{W_2^s(\mathbb{R}^d)}}\) with \(s\in \mathbb{R}\)
  • \(H_0^s(\Omega) = \overline{C_0^{\infty}(\Omega)}^{\norm{\cdot}_{H^s(\Omega)}}\)

Generate Sobolev spaces by taking restriction

  • \(H^s(\Omega) = \left\{ v=\widetilde{v}\vert_{\Omega}: \widetilde{v}\in H^s(\mathbb{R}^d) \right\}\)

Duality relation between Sobolev spaces

  • \(W_p^s(\Omega) = \left( \overset{\circ}{W}_q^{-s} \right)'\) with \(s < 0\) and \(p \in (1, \infty)\)
  • \(\overset{\circ}{W}_p^s(\Omega) = \left( W_q^{-s}(\Omega) \right)'\) with \(s < 0\) and \(p \in (1, \infty)\)
  • \(\widetilde{H}^s(\Omega) = \left[ H^{-s}(\Omega) \right]'\) for all \(s\in \mathbb{R}\) when \(\Omega\) is a Lipschitz domain
  • \(H^s(\Omega)=\left[ \widetilde{H}^{-s}(\Omega) \right]'\) for all \(s\in \mathbb{R}\) when \(\Omega\) is a Lipschitz domain
  • \(H^s(\Gamma)=\left[ H^{-s}(\Gamma) \right]'\) for \(s<0\)

Equivalent Sobolev spaces

  • \(H^s(\mathbb{R}^d) = W_2^s(\mathbb{R}^d)\) for all \(s\in \mathbb{R}\)
  • \(H^s(\Omega)=W_2^s(\Omega)\) for all \(s > 0\) when \(\Omega\) is a Lipschitz domain
  • \(\widetilde{H}^s(\Omega)=H_0^s(\Omega)\) for \(s\geq 0\) and \(s\neq \left\{ \frac{1}{2},\frac{3}{2},\frac{5}{2}, \cdots \right\}\), when \(\Omega\) is a Lipschitz domain