American Institute of Mathematical Sciences

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

A decomposition theorem for $BV$ functions

 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.
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.  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, (1977).   Google Scholar [5] P. Hajlasz and J. Malý, Approximation in Soblev spaces of nonlinear expressions involving the gradient,, Ark. Mat., 40 (2002), 245.  doi: 10.1007/BF02384536.  Google Scholar [6] J. J. Manfredi, Weakly monotone functions,, J. Geom. Anal., 4 (1994), 393.   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.  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, (1977).   Google Scholar [5] P. Hajlasz and J. Malý, Approximation in Soblev spaces of nonlinear expressions involving the gradient,, Ark. Mat., 40 (2002), 245.  doi: 10.1007/BF02384536.  Google Scholar [6] J. J. Manfredi, Weakly monotone functions,, J. Geom. Anal., 4 (1994), 393.   Google Scholar
 [1] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [2] Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098 [3] Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $(n, m)$-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117 [4] Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015 [5] Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106 [6] Dorothee Knees, Chiara Zanini. Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 121-149. doi: 10.3934/dcdss.2020332 [7] Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101 [8] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [9] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

2019 Impact Factor: 1.105