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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 $\Omega\subset \R^n$ 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 $Du\in\mathcal M(\Omega,\R^n)$ such that $$ \int_\Omega u(x)\,\mathrm{div}\phi(x)\, dx = - \int_\Omega \langle \phi(x), Du(x)\rangle $$ for every function $\phi\in C_c^1(\Omega,\R^n)$ . The measure $Du$ , represents the distributional derivative of $u$ since the above equality holds true for every $\phi\in C^\infty_c(\Omega,\R^n)$ .
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 $W^{1,1}$ .
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 $$ V(u,\Omega):=\sup\{\int_\Omega u\mathrm{div}\phi\colon \phi\in\mathcal C_c^1(\Omega,\R^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\}. $$ We define $BV(\Omega)=\{ u\in L^1(\Omega)\colon V(u,\Omega)<+\infty\}$ .
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"BV function" is owned by paolini.
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See Also: total variation
| Other names: |
function of bounded variation |
| Also defines: |
total variation |
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Cross-references: variation, equivalent, weak derivative, Lebesgue measure, equality, represents, measure, vector measure, open set, bounded variation, Radon measure, finite, derivative, functions
There are 4 references to this entry.
This is version 8 of BV function, born on 2005-04-27, modified 2005-05-01.
Object id is 6969, canonical name is BVFunction.
Accessed 20380 times total.
Classification:
| AMS MSC: | 26B30 (Real functions :: Functions of several variables :: Absolutely continuous functions, functions of bounded variation) |
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Pending Errata and Addenda
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