切向迹 (Tangential trace)

经典的切向迹

令 \(K\in \mathbb{R}^3\) 为多面体区域, 那么 Sobolev 空间 \(H(\mathrm{curl},K)\) 具有如下的迹定义 \(n\times \textbf{u}: H(\mathrm{curl},K)\to H^{-1/2}(\partial K)\): $$ \langle n\times \textbf{u}, \textbf{v}\rangle_{\partial K}:= (\nabla \times \textbf{u}, \textbf{v})-(\textbf{u}, \nabla \times \textbf{v}), \quad \forall \textbf{v}\in H^{1}(K)^3. $$

问题 1

那么, 对于 \(H(\mathrm{curl},K)\) 中的一般函数可以定义局部迹吗? 也就是说, 对任意面 \(F\in \partial K\), \((n\times \textbf{u})_{|F}\) 有意义吗, 或者在什么意义下是适定的?

答. \((n\times \textbf{u})_{|F}\in H^{-1/2}_{00}(F)\) 是适定的. 事实上, 对任意函数 \(\eta\in H^{1/2}_{00}(F)^3\), 其在 \(\partial K\) 上的零延拓 \(\tilde{\eta}\in H^{1/2}(\partial K)\); 另外, 由迹算子的逆可知总是存在 \(\bar{\tilde{\eta}}\in H^{1}(K)^3\) 使得 \(\bar{\tilde{\eta}}_{|\partial K}= \tilde{\eta}\), 并且满足 \[ \langle n\times \textbf{u}, \eta\rangle_{H^{-1/2}_{00}(F)^3}:= \langle n\times \textbf{u}, \tilde{\eta}\rangle_{\partial K}=(\nabla\times \textbf{u}, \bar{\tilde{\eta}})- (\textbf{u}, \nabla\times \bar{\tilde{\eta}}). \] 同时具有如下连续性估计 $$ \begin{equation*} \begin{split} \langle n\times \textbf{u}, \eta\rangle_{H^{-1/2}_{00}(F)^3} &\leq C (\Vert \textbf{u} \Vert_{L^2(K)}+\Vert \nabla \times\textbf{u} \Vert_{L^2(K)}) \Vert \tilde{\eta} \Vert_{H^{1/2}(\partial K)^3} \\ &\leq C (\Vert \textbf{u} \Vert_{L^2(K)}+\Vert \nabla \times\textbf{u} \Vert_{L^2(K)}) \Vert \eta \Vert_{H^{1/2}_{00}(F)^3} \end{split} \end{equation*} $$ 也就是说, 存在一个正数 \(C\) (依赖于区域 \(K\)) $$ \Vert n\times \textbf{u} \Vert_{H^{-1/2}_{00}(F)^3} \leq C (\Vert \textbf{u} \Vert_{L^2(K)}+\Vert \nabla \times\textbf{u} \Vert_{L^2(K)}). $$

问题 2

因为 \(C_0^{\infty}(F)\) 在 \(H^{1/2}(F)\) 中是稠密的, 那么对任意函数 \(\eta\in H^{1/2}(F)\) 且 \(\lim_{m\to \infty}\phi_m=\eta\), 是否可以定义如下切向迹 $$ \langle n\times \textbf{u}, \eta\rangle_{F}:= \lim_{m\to \infty} \langle n\times \textbf{u}, \phi_m\rangle_{H^{-1/2}_{00}(F)^3}? $$

答. 不行, 上述定义中, \(n\times \textbf{u}\) 不是 \(H^{1/2}(F)\) 上的连续线性泛函. 事实上, 由问题 1 中的定义可知 $$ \langle n\times \textbf{u}, \phi_m\rangle_{H^{-1/2}_{00}(F)^3} \leq C (\Vert \textbf{u} \Vert_{L^2(K)}+\Vert \nabla \times\textbf{u} \Vert_{L^2(K)}) \Vert \phi_m \Vert_{H^{1/2}_{00}(F)^3}, $$ 但是, 因为零延拓 \(\tilde{\eta}\notin H^{1/2}(\partial K)^3\), 所以并不存在一个常数 \(C\) 使得当 \(m\to \infty\) 时 $$ \Vert \phi_m \Vert_{H^{1/2}_{00}(F)^3} \leq C \Vert \phi_m \Vert_{H^{1/2}(F)^3}. $$ 事实上, 因为 \(\eta\in H^{1/2}(F)\), 通常我们会有 \(\eta\notin H^{1/2}_{00}(F)\), 即 \(\Vert \eta \Vert_{H^{1/2}_{00}(F)^3} = \infty \). 由 \(\Vert \eta-\phi_m \Vert_{H^{1/2}_{00}(F)^3}\to 0\) 可知 \(\Vert \phi_m \Vert_{H^{1/2}_{00}(F)^3} \to \infty \).