# BV function

###### Definition 1 (functions of bounded variation).

Let $\Omega\subset\mathbb{R}^{n}$ be an open set. We say that a function $u\in L^{1}(\Omega)$ has bounded variation  , and write $u\in BV(\Omega)$, if there exists a finite Radon vector measure $Du\in\mathcal{M}(\Omega,\mathbb{R}^{n})$ such that

 $\int_{\Omega}u(x)\,\mathrm{div}\phi(x)\,dx=-\int_{\Omega}\langle\phi(x),Du(x)\rangle$

for every function $\phi\in C_{c}^{1}(\Omega,\mathbb{R}^{n})$. The measure  $Du$, represents the distributional derivative of $u$ since the above equality holds true for every $\phi\in C^{\infty}_{c}(\Omega,\mathbb{R}^{n})$.

Notice that $W^{1,1}(\Omega)\subset BV(\Omega)$. In fact if $u\in W^{1,1}(\Omega)$ one can choose $\mu:=\nabla u\mathcal{L}$ (where $\mathcal{L}$ is the Lebesgue measure  on $\Omega$). The equality $\int u\mathrm{div\phi}=-\int\phi\,d\mu=-\int\phi\nabla u$ is nothing else than the definition of weak derivative, and hence holds true. One can easily find an example of a $BV$ functions which is not $W^{1,1}$.

###### Definition 2 (variation).

Given $u\in L^{1}(\Omega)$ we define the variation of $u$ in $\Omega$ as

 $V(u,\Omega):=\sup\{\int_{\Omega}u\mathrm{div}\phi\colon\phi\in\mathcal{C}_{c}^% {1}(\Omega,\mathbb{R}^{n}),\ \|\phi\|_{L^{\infty}(\Omega)}\leq 1\}.$

We define $BV(\Omega)=\{u\in L^{1}(\Omega)\colon V(u,\Omega)<+\infty\}$.

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Title BV function BVFunction 2013-03-22 15:12:32 2013-03-22 15:12:32 paolini (1187) paolini (1187) 11 paolini (1187) Definition msc 26B30 function of bounded variation TotalVariation total variation  