论文标题
Lipschitz域上Sobolev空间的微量运营商的内核
Kernel of Trace Operator of Sobolev Spaces on Lipschitz Domain
论文作者
论文摘要
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）$。可以在Nečas的专着\ Cite {Key-2}，定理4.10，§2.4.3中找到证明。 我们注意到这一点的主要结果是：我们通过证明两个封闭方式与跟踪操作员的内核相同，$ t：\，w^{1，p}（d）（d）\ rightArrow l^{p}（\ partial d）$通过evans和gariepy textboop的变量的一些变化来延长coite \ cite-cite formitation formination formination forrestion {在Evans广泛的PDE教科书\ cite {key-3}的第5.5节中，从$ \ Mathcal {C}^{1} $到Lipschitz域。
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 Nečas's monograph \cite{key-2}, Theorem 4.10, §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 §5.5 of Evans' widespread PDE textbook \cite{key-3}, from $\mathcal{C}^{1}$ to Lipschitz domain.
