Suppose \( s > 0 \), \( p \in [1, \infty) \). Then, the space \( W^{s,p}(\Omega) \) consists of functions \( u \in W^{m,p}(\Omega) \) with finite norm \( \|u\|_{W^{s,p}(\Omega)} \), where

\[ \|u\|_{W^{s,p}(\Omega)}^p := \|u\|_{W^{m,p}(\Omega)}^p + \sum_{|\alpha|=m} [D^\alpha u]_{W^{\sigma,p}(\Omega)}^p \]

with \( s = m + \sigma \), \( m \in \mathbb{N}_0 \), \( \sigma \in (0,1) \) and

\[ [u]_{W^{\sigma,p}(\Omega)}^p = \int_\Omega \int_\Omega \frac{|u(x) - u(y)|^p}{|x - y|^{d + \sigma p}} \, dx \, dy. \]
Remarks
  • \( W^{s,\infty }(\Omega) \) corresponds to \( C^{s,\infty }(\overline{\Omega})\) [1]. That is why, we assume \(p\neq \infty\).
  • \(W^{s,p}(\Omega)\) with the \(\lVert \cdot\rVert_{W^{s,p}(\Omega)}\) norm is a Banach space [1, Satz 6.32].
  • The norm \(\lVert \cdot\rVert_{W^{s,2}(\Omega)}\) is not continuous in \(s=0\), i.e. there are functions \(u\) with \[ \lim_{s \to 0}\lVert u\rVert_{W^{s,2}(\Omega)}\neq \lVert u\rVert_{L^2(\Omega)} \] [1, Beispiel 6.33].
  • The function space \( C^\infty(\overline{\Omega}) \cap W^{s,p}(\Omega) \) is dense in \( W^{s,p}(\Omega) \) [1, Satz 6.37].
  • If \(\Omega\) is a bounded Lipschitz domain , then the restriction of \( C^\infty_c(\mathbb{R}^d) \) on \(\Omega\) is dense in \( H^{s,p}(\Omega) \) [1, Satz 6.37].

See also Link to heading

References Link to heading

  1. M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. doi:10.1007/978-3-642-15269-6