BV function
Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure
. More precisely:
Definition 1 (functions of bounded variation).
Let Ω⊂Rn be an open set. We say that a function u∈L1(Ω)
has bounded variation, and write u∈BV(Ω), if there exists a finite Radon vector measure Du∈M(Ω,Rn) such that
∫Ωu(x)divϕ(x)𝑑x=-∫Ω⟨ϕ(x),Du(x)⟩ |
for every function ϕ∈C1c(Ω,Rn). The measure Du,
represents the distributional derivative of u since the above equality holds true for every ϕ∈C∞c(Ω,Rn).
Notice that W1,1(Ω)⊂BV(Ω). In fact if u∈W1,1(Ω) one can choose μ:= (where is the Lebesgue measure on ).
The equality
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 .
An equivalent definition can be given as follows.
Definition 2 (variation).
Given we define the variation of in as
We define .
.
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![]() |