﻿ N维空间中弱导数的一些性质

# N维空间中弱导数的一些性质Some Properties of Weak Derivative in N-Dimensional Space

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

2. 弱导数的定义

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

${\int }_{\Omega }{u}_{{x}_{i}}\upsilon \text{d}x=-{\int }_{\Omega }u{\upsilon }_{{x}_{i}}\text{d}x+{\int }_{\partial \Omega }u\upsilon {\nu }^{i}\text{d}S\text{\hspace{0.17em}}\left(i=1,\cdots ,n\right)$.

3. 弱导数的性质

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

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

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

1) 对 $\forall \varphi \in {C}_{0}^{\infty }\left(\Omega \right)$，有 $\underset{j\to \infty }{\mathrm{lim}}{\int }_{\Omega }{v}_{j}\varphi \text{d}x={\int }_{\Omega }v\varphi \text{d}x$

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

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

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

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

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

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

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

$\therefore$ 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 ${C}_{\beta }^{\sigma -1}+{C}_{\beta }^{\sigma }={C}_{\beta +1}^{\sigma }$

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

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

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

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

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

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

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

4. 结束语

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[4] 王跃, 索洪敏, 吴徳科, 彭林艳, 蔡梅梅. 弱导数与弱解的一个注记[J]. 湖北民族大学学报(自然科学版), 2020, 38(1): 58-63.

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