BV function

Functions of bounded variation, $B V$ 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

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

Title	BV function
Canonical name	BVFunction
Date of creation	2013-03-22 15:12:32
Last modified on	2013-03-22 15:12:32
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	11
Author	paolini (1187)
Entry type	Definition
Classification	msc 26B30
Synonym	function of bounded variation
Related topic	TotalVariation
Defines	total variation