Kernel of Trace Operator of Sobolev Spaces on Lipschitz Domain

Abstract

We are going to show that on bounded Lipschitz domain $D$: both $C_{c}^{\infty}(D)$, the set of smooth functions on $D$ with compact support, and $C_{0}^{\infty}(D)$, the set of smooth functions on $D$ with (extension) zero boundary, are dense in $W^{1,p}\left(D\right)$, $p\in[1,\infty)$. A proof can be found in Ne\v{c}as's monograph \cite{key-2}, Theorem 4.10, {\S}2.4.3. Our main result in this note is that: we find another proof by showing that both closures is the same as kernel of trace operator $T:\,W^{1,p}(D)\rightarrow L^{p}(\partial D)$ via some change of variables formulas from Evans and Gariepy's textbook \cite{key-4} for Lipschitz coordinate transformation, to extend the proof of Theorem 2 in {\S}5.5 of Evans' widespread PDE textbook \cite{key-3}, from $\mathcal{C}^{1}$ to Lipschitz domain.