Assume \(U\subset \mathbb{R}^n\) is bounded, \(\partial U\) is \(C^1\) and \(p\in [1,\infty)\). Then there exists a bounded operator
\begin{equation*} T\colon W^{1,p}(U)\to L^p(\partial U) \end{equation*}such that
- \(Tu=u|_{\partial U}\) if \(u\in W^{1,p}(U)\cap C^b(U)\)
for every \(u\in W^{1,p}(U)\) and with the constant \(C>0\) depending only on \(p\) and \(U\). We call \(Tu\) the trace of \(u\) on \(\partial U\). [1, 5.5 Theorem 1]
Remarks
- \(u\in W^{1,p}_0(U)\) if and only if \(Tu=0\) on \(\partial U\) (see (0x674233bd) ).
- For \(p>1\), the trace of Sobolev functions habe sligtly better regularity then \(L^p(\partial U)\). However, for \(p=1\) this is not the case (see (0x68e62a05) ).
- The trace theorem remains true for bounded Lipschitz domains [2, Satz 6.15].
- For \(p=\infty\). Since \(W^{1,\infty }(U)\) functions a continuous, the trace oprator is a classical one and it is bounded as map from \(C(\partial U)\) to \(W^{1,\infty }(U)\) (see (0x68e62b3c) ).
See also Link to heading
- Sobolev trace type inequality on a cylinder
- trace theorem on \(\mathbb{R}^d_+\)
- trace theorem on subsets of \(\partial\mathbb{R}^{d}_+\)
References Link to heading
- L. Evans, Partial differential equations. Providence (R. I.): American mathematical society, 1998.
- M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. doi:10.1007/978-3-642-15269-6