# 总变差

## 定义

### 矢量空间

${\displaystyle V_{b}^{a}(f)=\int _{a}^{b}|f'(x)|\,dx.}$

${\displaystyle V_{b}^{a}(f)=\sup _{P}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,\,}$

${\displaystyle V(f,\Omega ):=\sup \left\{\int _{\Omega }f\mathrm {div} \varphi \colon \varphi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert \varphi \Vert _{L^{\infty }(\Omega )}\leq 1\right\},}$

${\displaystyle V(f,\Omega )=\int \limits _{\Omega }\left|\nabla f\right|.}$

### 度量空间

${\displaystyle |\mu |(E)=\sup _{\pi }\sum _{A\in \pi }|\mu (A)|\qquad \forall E\in \Sigma }$

${\displaystyle |\mu |=\mu ^{+}+\mu ^{-}\,}$

## 可微定义的证明

### 引理

${\displaystyle \int \limits _{\Omega }f\,\mathrm {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }$

#### 引理证明

${\displaystyle \int \limits _{\Omega }{\text{div}}\left(f\mathbf {\varphi } \right)=\int \limits _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} }$

${\displaystyle \int \limits _{\Omega }{\text{div}}\left(f\mathbf {\varphi } \right)=\int \limits _{\partial \Omega }\left(f\mathbf {\varphi } \right)\cdot \mathbf {n} =0}$

${\displaystyle \int \limits _{\Omega }f\,\mathrm {div} \varphi =-\int _{\Omega }\nabla f\cdot \varphi }$.