Let \(s\ge 0\) and \(\Omega\subseteq \mathbb{R}^d\) be a domain. The fractional Sobolev space \(W^{s,2}(\Omega)\) is a Hilbert space , and it is denoted by \(H^s(\Omega)\). The inner product is given by

\[ \langle u, v\rangle_{H^s(\Omega)}=\langle u, v\rangle_{H^k(\Omega)} + \sum_{\lvert \alpha\rvert=k}\langle D^\alpha u, D^\alpha v\rangle_\sigma \]

with \(s=k+\sigma\), \(\sigma< 1\) and

\[ \langle u, v\rangle_\sigma = \int_{\Omega} \int_{\Omega} \frac{(u(x)-u(y))\overline{(v(x)-v(y))}}{\lvert x-y\rvert^{d+2\sigma}} \,dx \,dy \]

[1, Satz 6.32].

Inequalities Link to heading

  • \([u]_{H^{1/2}(\mathbb{R}^d)}\) vs. \([u]_{H^{1/2}(\Omega)}\) (see (0x68cbf31f) ),
  • inequality for \([a\cdot u]_{H^{1/2}(\Omega)}\) (see (0x68cbf438) )

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