N维空间中弱导数的一些性质

Some Properties of Weak Derivative in N-Dimensional Space

摘要: 针对某些不满足可导条件的函数，充分运用变分法基本引理、分部积分公式等方法来推广弱导数的性质。从一维的弱导数性质出发，总结证明N维空间中弱导数的一些常用性质，深化分部积分在定积分中的实际应用。通过偏微分方程的学习，N维空间中弱导数的性质更适用于解决高维问题。

Abstract: For some functions that do not satisfy the differentiable condition, the basic lemma of variational method and partial integral formula are fully used to generalize the properties of weak derivative. Starting from the property of one-dimensional weak derivative, this paper summarizes the property of weak derivative in dimensional space, and deepens the practical application of partial integral in definite integral. Through the learning of partial differential equations, the properties of weak derivatives in N-dimensional space are more suitable for solving higher dimensional problems.

1. 引言

在偏微分方程的学习中，数 [1] [2] 知识几乎贯穿整个数学体系。由于导数条件的限制与实际情况的需要，导致N维弱导数 [3] [4] [5] [6] 出现。弱导数是一个函数的微分(强微分)概念的推广。通过弱导数和磨光函数将函数磨光，在函数没有可微的条件下，也能得到和函数可导相似的结果。本文主要证明了N维空间中弱导数的一些性质。该结果贯穿于整个偏微分方程的学习中，特别是解决了光滑函数的局部逼近和全局逼近问题。

2. 弱导数的定义

引理1 [5] [6]：设 $u,\upsilon \in L_{weak}^1\left(\Omega \right)$， $\alpha$ 是多重指标，若任一测试函数 $\varphi \in C_c^{\infty }\left(\Omega \right)$ 满足 ${\displaystyle {\int }_{\Omega }u{D}^{\alpha }\varphi \text{d}x}={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }\upsilon \varphi \text{d}x}$，则称 $\upsilon$ 是u的 $\alpha$ 阶弱导数，记作 $D^{\alpha }u=\upsilon$。

引理2 [5]：设函数 $\varphi \left(x\right)$ 定义在开区间 $I=\left(-1,1\right)$ 上，若 $\varphi \left(x\right)$ 在I上无穷次可微，且u的支集 $spt\left\{\varphi \right\}=\left\{\overline{x\in I:\varphi \left(x\right)\ne 0}\right\}\subset \subset I$，则称 $\varphi \left(x\right)$ 是试验函数，记为 $\varphi \in C_c^{\infty }\left(I\right)$。

引理3 [7]：(变分法基本引理) 若 $u\in C^0\left(\Omega \right)$ 且对任意的 $\varphi \in C_c^{\infty }\left(\Omega \right)$ 有

${\displaystyle {\int }_{\Omega }u\left(x\right)\varphi \left(x\right)\text{d}x=0}$，则在 $\Omega$ 中 $u\equiv 0$。

引理4 [6]：(分部积分公式) 设 $u,\upsilon \in C^1\left(\bar{\Omega }\right)$，则

${\displaystyle {\int }_{\Omega }{u}_{x_i}}\upsilon \text{d}x=-{\displaystyle {\int }_{\Omega }u}{\upsilon }_{x_i}\text{d}x+{\displaystyle {\int }_{\partial \Omega }u}\upsilon {\nu }^i\text{d}S\left(i=1,\cdots ,n\right)$.

3. 弱导数的性质

性质1：(唯一性)若u的 $\alpha$ 阶弱导数存在，则在零测集的意义下一定唯一。

证明：假设 $\upsilon ,\bar{\upsilon }\in L_{loc}^1\left(\Omega \right)$ 都是u的 $\alpha$ 阶弱导数，从而对 $\forall \varphi \in C_c^{\infty }\left(\Omega \right)$，有

${\displaystyle {\int }_{\Omega }u{D}^{\alpha }}\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }\upsilon }\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }\bar{\upsilon }}\varphi \text{d}x$,

即对 $\forall \varphi \in C_c^{\infty }\left(\Omega \right)$，有 ${\displaystyle {\int }_{\Omega }\left(\upsilon -\bar{\upsilon }\right)\varphi \text{d}x}=0$。

由此可知， $\upsilon =\bar{\upsilon }$ 在 $\Omega$ 上几乎处处成立，从而原求证成立。

性质2：设 $\alpha$ 和 $\beta$ 是两个多重指标，如果 $D^{\alpha +\beta }u$ 存在，那么 $D^{\beta }\left(D^{\alpha }u\right)$ 和 $D^{\alpha }\left(D^{\beta }u\right)$ 存在，并且 $D^{\beta }\left(D^{\alpha }u\right)=D^{\alpha }\left(D^{\beta }u\right)=D^{\alpha +\beta }u$。

证明：固定 $\varphi \in C_0^{\infty }\left(\Omega \right)$，则 $D^{\beta }\varphi \in C_c^{\infty }\left(\Omega \right)$，根据引理1有

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{D}^{\alpha }u}{D}^{\beta }\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }u{D}^{\alpha +\beta }}\varphi \text{d}x\\ ={\left(-1\right)}^{\left|\alpha \right|}{\left(-1\right)}^{\left|\alpha +\beta \right|}{\displaystyle {\int }_{\Omega }{D}^{\alpha +\beta }}u\cdot \varphi \text{d}x\\ ={\left(-1\right)}^{\left|\beta \right|}{\displaystyle {\int }_{\Omega }{D}^{\alpha +\beta }}u\cdot \varphi \text{d}x.\end{array}$

而 ${\displaystyle {\int }_{\Omega }{D}^{\alpha }}u{D}^{\beta }\varphi \text{d}x={\left(-1\right)}^{\left|\beta \right|}{\displaystyle {\int }_{\Omega }{D}^{\beta }}\left(D^{\alpha }u\right)\cdot \varphi \text{d}x$，

从而有 ${\displaystyle {\int }_{\Omega }{D}^{\alpha +\beta }}u\cdot \varphi \text{d}x={\displaystyle {\int }_{\Omega }{D}^{\beta }}\left(D^{\alpha }u\right)\cdot \varphi \text{d}x$。

因此 $D^{\beta }\left(D^{\alpha }u\right)=D^{\alpha +\beta }u$。

同理有 $D^{\alpha }\left(D^{\beta }u\right)=D^{\alpha +\beta }u$。

故在弱意义下 $D^{\beta }\left(D^{\alpha }u\right)$， $D^{\alpha }\left(D^{\beta }u\right)$ 存在，且等于 $D^{\alpha +\beta }u$。

性质3：设 $1<p<\infty$， $v\in L^p\left(\Omega \right)$，假设存在函数列 $\left\{v_j\right\}$ 满足 $v_j,D{v}_j\in L^p\left(\Omega \right)$ 以及

1) 对 $\forall \varphi \in C_0^{\infty }\left(\Omega \right)$，有 $\underset{j\to \infty }{\lim }{\displaystyle {\int }_{\Omega }{v}_j}\varphi \text{d}x={\displaystyle {\int }_{\Omega }v}\varphi \text{d}x$ ；

2) 存在正常数C，使得 ${\Vert D{v}_j\Vert }_{L^p\left(\Omega \right)}\le C$， $\forall j\in N^{+}$。

那么v在 $\Omega$ 内存在一阶弱导数 $Dv$，并且 $Dv\in L^p\left(\Omega \right)$， ${\Vert Dv\Vert }_{L^p\left(\Omega \right)}\le C$。

证明：由(2)知存在 $\left\{D{v}_j\right\}$ 的子列，不妨设为 $\left\{D{v}_j\right\}$ 以及 $u\in L^p\left(\Omega \right)$ 在 $L^p\left(\Omega \right)$ 中 $D{v}_j\rightharpoonup u$ 且 ${\Vert u\Vert }_{L^p\left(\Omega \right)}\le C$。显然，对 $\forall \varphi \in C_0^{\infty }\left(\Omega \right)$ 有

$\underset{j\to \infty }{\lim }{\displaystyle {\int }_{\Omega }\varphi D{v}_j}\text{d}x={\displaystyle {\int }_{\Omega }\varphi u}\text{d}x$,

由弱导数的定义知

${\displaystyle {\int }_{\Omega }\varphi D{v}_j}\text{d}x=-{\displaystyle {\int }_{\Omega }{v}_{j}D\varphi \text{d}x}$,

结合1)我们有

${\displaystyle {\int }_{\Omega }\varphi u\text{d}x=-{\displaystyle {\int }_{\Omega }vD\varphi \text{d}x}}$,

故 $u=Dv\in L^p\left(\Omega \right)$，并且 ${\Vert Dv\Vert }_{L^p\left(\Omega \right)}\le C$。

从而原求证成立。

性质4：假设 $u,v\in W^{k,p}\left(\Omega \right)$， $\left|\alpha \le k\right|$，那么对每个 $\lambda ,\mu \in R$， $\lambda u+\mu v\in W^{k,p}\left(\Omega \right)$ 并且当 $\left|\alpha \le k\right|$ 时有 $D^{\alpha }\left(\lambda u+\mu v\right)=\lambda D^{\alpha }u+\mu D^{\alpha }v$。

证明：固定 $\varphi \in C_0^{\infty }\left(\Omega \right)$，则 $D^{\alpha }\varphi \in C_c^{\infty }\left(\Omega \right)$。

$\begin{array}{c}{\displaystyle {\int }_{\Omega }{D}^{\alpha }}\left(\lambda u+\mu v\right)\varphi \text{d}x={\displaystyle {\int }_{\Omega }{D}^{\alpha }}\lambda u\varphi \text{d}x+{\displaystyle {\int }_{\Omega }{D}^{\alpha }}\mu v\varphi \text{d}x\\ ={\displaystyle {\int }_{\Omega }\lambda D^{\alpha }}u\varphi \text{d}x+{\displaystyle {\int }_{\Omega }\mu D^{\alpha }}v\varphi \text{d}x\\ ={\displaystyle {\int }_{\Omega }\left(\lambda D^{\alpha }u+\mu D^{\alpha }v\right)\varphi \text{d}x}.\end{array}$

因此 $D^{\alpha }\left(\lambda u+\mu v\right)=\lambda D^{\alpha }u+\mu D^{\alpha }v$。

从而原求证成立。

性质5：假设 $u,v\in W^{k,p}\left(\Omega \right)$， $\left|\alpha \le k\right|$，如果 $\xi \in C_c^{\infty }\left(\Omega \right)$，那么 $\xi u\in W^{k,p}\left(\Omega \right)$ 且 $D^{\alpha }\left(\xi u\right)={\displaystyle \underset{\beta \le \alpha }{\sum }\left(\begin{array}{c}\beta \\ \alpha \end{array}\right)D^{\beta }\xi D^{\alpha -\beta }u}$，其中 $\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)=\frac{\alpha !}{\beta !\left(\alpha -\beta \right)!}$。

证明：先证明 $\left(\begin{array}{c}\beta \\ \sigma -\gamma \end{array}\right)+\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)=\left(\begin{array}{c}\alpha \\ \sigma \end{array}\right)$。

$\because \left(\begin{array}{c}\alpha \\ \beta \end{array}\right)=\frac{\alpha !}{\beta !\left(\alpha -\beta \right)!}=C_{\alpha }^{\beta }$,

$\therefore$ 我们将其当作排 ${\left(uv\right)}^{\prime }=u^{\prime }v+u{v}^{\prime }$ 列组合来证明 $C_{\beta }^{\sigma -\gamma }+C_{\beta }^{\sigma }=C_{\alpha }^{\sigma }=C_{\beta +\gamma }^{\sigma }$，其中 $\gamma =1$，即证明 $C_{\beta }^{\sigma -1}+C_{\beta }^{\sigma }=C_{\beta +1}^{\sigma }$。

$\begin{array}{c}{C}_{\beta }^{\sigma -1}+C_{\beta }^{\sigma }=\frac{\beta !}{\left(\sigma -1\right)!\left(\beta +1-\sigma \right)!}+\frac{\beta !}{\sigma !\left(\beta -\sigma \right)!}\\ =\frac{\beta !}{\left(\sigma -1\right)!\left(\beta +1-\sigma \right)!}+\frac{\beta !\left(\beta -\sigma +1\right)}{\sigma !\left(\beta -\sigma +1\right)!}\\ =\frac{\beta !(\beta -\sigma +1+\sigma )}{\sigma !\left(\beta -\sigma +1\right)!}=\frac{\beta !\left(\beta +1\right)}{\sigma !\left(\beta -\sigma +1\right)!}\\ =\frac{\left(\beta +1\right)!}{\sigma !\left(\beta -\sigma +1\right)!}=C_{\beta +1}^{\sigma }.\end{array}$

下面我们用数学归纳法来证明：若u有k阶弱导，则 $\xi u$ 也有k阶弱导。

第一步 假设 $\left|\alpha \right|=1$，任一 $\varphi \in C_0^{\infty }\left(\Omega \right)$，

则由分部积分公式有 ${\left(uv\right)}^{\prime }=u^{\prime }v+u{v}^{\prime }$

得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }\xi u{D}^{\alpha }}\varphi \text{d}x={\displaystyle {\int }_{\Omega }\left[u{D}^{\alpha }\left(\xi \varphi \right)-u\left(D^{\alpha }\xi \right)\varphi \right]\text{d}x}\\ ={\displaystyle {\int }_{\Omega }u{D}^{\alpha }}\left(\xi \varphi \right)\text{d}x-{\displaystyle {\int }_{\Omega }u\left(D^{\alpha }\xi \right)\varphi \text{d}x}\\ ={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }\left(D^{\alpha }u\right)\xi \varphi \text{d}x}-{\displaystyle {\int }_{\Omega }u\left(D^{\alpha }\xi \right)\varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\left(D^{\alpha }u\right)\xi \varphi \text{d}x}-{\displaystyle {\int }_{\Omega }u\left(D^{\alpha }\xi \right)\varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\xi \left(D^{\alpha }u\right)\varphi \text{d}x}-{\displaystyle {\int }_{\Omega }u\left(D^{\alpha }\xi \right)\varphi \text{d}x}\\ =-{\displaystyle {\int }_{\Omega }\left[\xi \left(D^{\alpha }u\right)+u\left(D^{\alpha }\xi \right)\right]\varphi \text{d}x}.\end{array}$

而 ${\displaystyle {\int }_{\Omega }\xi u{D}^{\alpha }}\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }{D}^{\alpha }\left(\xi u\right)\varphi \text{d}x}=-{\displaystyle {\int }_{\Omega }{D}^{\alpha }\left(\xi u\right)\varphi \text{d}x}$。

因此 $D^{\alpha }\left(\xi u\right)=\xi D^{\alpha }u+u{D}^{\alpha }\xi$。

第二步 现在假设对 $\left|\alpha \right|\le l$ 成立的所有函数 $\xi$，原求证都成立。

第三步 当 $\left|\alpha \right|=l+1$ 时，对某个 $\left|\beta \right|=l,\left|\gamma \right|=1,\alpha =\beta +l$，上述的 $\varphi \in C_0^{\infty }\left(\Omega \right)$ 有

$\begin{array}{c}{\displaystyle {\int }_{\Omega }\xi u{D}^{\alpha }}\varphi \text{d}x={\displaystyle {\int }_{\Omega }\xi u{D}^{\beta +\gamma }}\varphi \text{d}x\\ ={\displaystyle {\int }_{\Omega }\xi u{D}^{\beta }\left(D^{\gamma }\varphi \right)\text{d}x}\\ ={\left(-1\right)}^{\left|\beta \right|}{\displaystyle {\int }_{\Omega }{D}^{\beta }}\left(\xi u\right)D^{\gamma }\varphi \text{d}x\\ ={\left(-1\right)}^{\left|\beta \right|}{\displaystyle {\int }_{\Omega }{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\sigma }\xi D^{\beta \text{-}\sigma }u{D}^{\gamma }\varphi \text{d}x}}\\ ={\left(-1\right)}^{\left|\beta \right|+\left|\gamma \right|}{\displaystyle {\int }_{\Omega }{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\gamma }}}\left(D^{\sigma }\xi D^{\beta \text{-}\sigma }u\right)\varphi \text{d}x,\end{array}$

代入 $D^{\alpha }\left(\xi u\right)=\xi D^{\alpha }u+u{D}^{\alpha }\xi$，

得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }\xi u{D}^{\alpha }}\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)}}\left[D^{\sigma }\xi D^{\gamma }{D}^{\beta -\sigma }u+D^{\sigma }\xi D^{\alpha -\sigma }u\right]\varphi \text{d}x\\ ={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)}}\left[D^{\sigma +\gamma }\xi D^{\alpha -\left(\sigma +\gamma \right)}u+D^{\sigma }\xi D^{\alpha -\sigma }u\right]\varphi \text{d}x\\ ={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }\left[{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\sigma +\gamma }\xi D^{\alpha -\left(\sigma +\gamma \right)}u+}{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\sigma }\xi D^{\alpha -\sigma }u\varphi }\right]\text{d}x},\end{array}$ (1)

对于 ${\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\sigma +\gamma }\xi D^{\alpha -\left(\sigma +\gamma \right)}u}$，令 $\rho =\sigma +\gamma$ 则 $\beta =\alpha +\sigma -\rho$。

从而

$\begin{array}{c}{\displaystyle \underset{\sigma \le \beta }{\sum }\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)D^{\sigma +\gamma }\xi D^{\alpha -\left(\sigma +\gamma \right)}u}={\displaystyle \underset{\sigma \le \alpha +\sigma -\rho }{\sum }\left(\begin{array}{c}\beta \\ \rho -\gamma \end{array}\right)D^{\rho }\xi D^{\alpha -\rho }u}\\ ={\displaystyle \underset{\rho \le \alpha }{\sum }\left(\begin{array}{c}\beta \\ \rho -\gamma \end{array}\right)D^{\rho }\xi D^{\alpha -\rho }u}\\ \stackrel{\rho ,\sigma 的任意性}{=}{\displaystyle \underset{\sigma \le \alpha }{\sum }\left(\begin{array}{c}\beta \\ \sigma -\gamma \end{array}\right)D^{\sigma }\xi D^{\alpha -\sigma }u}.\end{array}$ (2)

将(2)代入(1)得

$\begin{array}{c}{\displaystyle {\int }_{\Omega }\xi }u{D}^{\alpha }\varphi \text{d}x={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }{\displaystyle \underset{\sigma \le \alpha }{\sum }\left[\left(\begin{array}{c}\beta \\ \sigma -\gamma \end{array}\right)+\left(\begin{array}{c}\beta \\ \sigma \end{array}\right)\right]D^{\sigma }\xi D^{\alpha -\sigma }u\varphi \text{d}x}}\\ ={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle \underset{\sigma \le \alpha }{\sum }\left(\begin{array}{c}\alpha \\ \sigma \end{array}\right)D^{\sigma }\xi D^{\alpha -\sigma }u\varphi \text{d}x}\\ ={\left(-1\right)}^{\left|\alpha \right|}{\displaystyle {\int }_{\Omega }{D}^{\alpha }\left(\xi u\right)\varphi \text{d}x}.\end{array}$

因此，当 $\left|\alpha \right|=l+1$ 时， $\xi u$ 也n有 $l+1$ 阶弱导。

综上所述，原求证成立。

4. 结束语

本文主要是对N维空间中弱导数的性质进行总结证明，将一维中的弱导数推广到N维空间中，从而得到比一维空间中弱导数更广泛的性质。根据本文推广的形式，后续可以继续改进推广更高阶线性形式。