Let \(\Omega\subset \mathbb{R}^n\) be a domain, \(k\in \mathbb{N}\) and \(p\in [1,\infty]\). The Sobolev space \(W^{k,p}(\Omega)\) contains all functions with weak derivatives in \(L^p(\Omega)\). For convenience we define \(W^{0,p}(\Omega)=L^p(\Omega)\).
The map
\begin{equation*} \lVert f\rVert_{W^{k,p}(\Omega)}^p=\sum_{\lvert \alpha\rvert\le k} \lVert D^\alpha f\rVert_{L^p(\Omega)}^p, \end{equation*}defines a norm on \(W^{k,p}(\Omega)\).
Remarks
Related Spaces Link to heading
- \(H^k(\Omega)\)
- \(W^{1,p}_0(\Omega)\)
- \(W^{1,p}_{\text{loc}}(\Omega)\)
- fractional Sobolev space \(W^{s,p}(\Omega)\)