A decomposition theorem for $BV$ functions

Stefano Bianchini; Daniela Tonon

doi:10.3934/cpaa.2011.10.1549

Article Contents

2011, Volume 10, Issue 6: 1549-1566. Doi: 10.3934/cpaa.2011.10.1549

This issue Previous Article Next Article Characterization of the value function of final state constrained control problems with BV trajectories

A decomposition theorem for $BV$ functions

Stefano Bianchini^1, and
Daniela Tonon^1,

1.
SISSA, via Bonomea, 265, Trieste, 34136

Received: April 30, 2010

Revised: February 28, 2011

Published: October 2011

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Abstract

The Jordan decomposition states that a function $f: R\to R$ is of bounded variation if and only if it can be written as the difference of two monotone increasing functions.
In this paper we generalize this property to real valued $BV$ functions of many variables, extending naturally the concept of monotone function. Our result is an extension of a result obtained by Alberti, Bianchini and Crippa.
A counterexample is given which prevents further extensions.

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Mathematics Subject Classification: Primary: 26B30, 26B35; Secondary: 28A75.

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References

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