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November  2011, 10(6): 1549-1566. doi: 10.3934/cpaa.2011.10.1549

A decomposition theorem for $BV$ functions

Stefano Bianchini 1, and  Daniela Tonon 1,

1. 

SISSA, via Bonomea, 265, Trieste, 34136, Italy, Italy

Received  May 2010 Revised  March 2011 Published  May 2011

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.
Keywords: Jordan decomposition., $BV$ functions, Monotone functions.
Mathematics Subject Classification: Primary: 26B30, 26B35; Secondary: 28A7.
Citation: Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Invariants for weakly regular ODE flows,, to appear., ().   Google Scholar

[2]

L. Ambrosio, V. Caselles, S. Masnou and J. M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS), 3 (2001), 39-92. doi: 10.1007/PL00011302.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2000.  Google Scholar

[4]

R. Engelking, "General Topology," PWN, Warsaw, 1977.  Google Scholar

[5]

P. Hajlasz and J. Malý, Approximation in Soblev spaces of nonlinear expressions involving the gradient, Ark. Mat., 40 (2002), 245-274. doi: 10.1007/BF02384536.  Google Scholar

[6]

J. J. Manfredi, Weakly monotone functions, J. Geom. Anal., 4 (1994), 393-402.  Google Scholar

show all references

References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Invariants for weakly regular ODE flows,, to appear., ().   Google Scholar

[2]

L. Ambrosio, V. Caselles, S. Masnou and J. M. Morel, Connected components of sets of finite perimeter and applications to image processing, J. Eur. Math. Soc. (JEMS), 3 (2001), 39-92. doi: 10.1007/PL00011302.  Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford University Press, 2000.  Google Scholar

[4]

R. Engelking, "General Topology," PWN, Warsaw, 1977.  Google Scholar

[5]

P. Hajlasz and J. Malý, Approximation in Soblev spaces of nonlinear expressions involving the gradient, Ark. Mat., 40 (2002), 245-274. doi: 10.1007/BF02384536.  Google Scholar

[6]

J. J. Manfredi, Weakly monotone functions, J. Geom. Anal., 4 (1994), 393-402.  Google Scholar

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