Kernel of Trace Operator of Sobolev Spaces on Lipschitz Domain

I-Shing Hu

Published 2020-04-29Version 1

We are going to show that \[ \overline{C_0^\infty \left(D\right)} = \overline{C_c^\infty \left(D\right)} \] in $W^{1,p}\left(D\right)$, $p\in[1,\infty)$, on Lipschitz domain $D$ by showing both sides are kernel of trace operator \[ T:\,W^{1,p}(D)\rightarrow L^{p}(\partial D). \] In Grisvard's book \cite{key-1}, Corollary 1.5.1.6 states a much more general result which covers above. But we cannot find a complete proof in literature. Fortunately, we apply some change of variables formulas from Evans and Gariepy's book (Theorem 3.9 and 3.11, \cite{key-3}), for Lipschitz coordinate transformation, in to improve the proof of \[ \overline{C_0^\infty \left(D\right)} = \overline{C_c^\infty \left(D\right)} \] in $W^{1,p}\left(D\right)$, $p\in[1,\infty)$, from $\mathcal{C}^{1}$ (Theorem 2 in {\S}5.5 of Evans' book \cite{key-2}) to Lipschitz domain.