PlanetMath: BV function

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BV function

(Definition)

Functions of bounded variation, functions, are functions whose distributional derivative is a finite Radon measure. More precisely:

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

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

for every function $\phi\in C_c^1(\Omega,\mathbb{R}^n)$ . The measure , represents the distributional derivative of 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 functions which is not $W^{1,1}$ .

An equivalent definition can be given as follows.

Definition 2 (variation) Given $u\in L^1(\Omega)$ we define the variation of in $\Omega$ as

$\displaystyle V(u,\Omega):=\sup\{\int_\Omega u\mathrm{div}\phi\colon \phi\in\mathcal C_c^1(\Omega,\mathbb{R}^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\}.$

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

"BV function" is owned by paolini.

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