BV function

Functions of bounded variation, BV functions, are functions whose distributional derivativePlanetmathPlanetmath is a finite Radon measureMathworldPlanetmath. More precisely:

Definition 1 (functions of bounded variation).

Let ΩRn be an open set. We say that a function uL1(Ω) has bounded variationMathworldPlanetmath, and write uBV(Ω), if there exists a finite Radon vector measure DuM(Ω,Rn) such that


for every function ϕCc1(Ω,Rn). The measureMathworldPlanetmath Du, represents the distributional derivative of u since the above equality holds true for every ϕCc(Ω,Rn).

Notice that W1,1(Ω)BV(Ω). In fact if uW1,1(Ω) one can choose μ:=u (where is the Lebesgue measureMathworldPlanetmath on Ω). The equality udivϕ=-ϕ𝑑μ=-ϕu 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 W1,1.

An equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition can be given as follows.

Definition 2 (variation).

Given uL1(Ω) we define the variation of u in Ω as


We define BV(Ω)={uL1(Ω):V(u,Ω)<+}.


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 variationMathworldPlanetmath