Assume \(U\subset \mathbb{R}^n\) is bounded, \(\partial U\) is \(C^1\) and \(p\in [1,\infty]\). Select a bounded open set \(V\) such that \(U\subset \subset V\) (i.e. \(U\subset \bar{U}\subset V)\). Then there is a bounded operator
\begin{equation*} E\colon W^{1,p}(U)\to W^{1,p}(\mathbb{R}^n), \end{equation*}such that
- \(Eu=u\) a.e. in \(U\),
- \(\supp Eu \subset V\),
for every \(u\in W^{k,p}(U)\) and with the constant \(C>0\) depending only on \(p\), \(U\) and \(V\).
The operator \(E\) is called extension of \(u\) to \(\mathbb{R}^n\). [1, 5.4 Theorem 1]
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.