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Tonelli plane variation

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2010 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A45 [MSN][ZBL]

A generalization to functions of two variables of the Variation of a function of one variable, proposed by Tonelli in [To].

Definition Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as \[ V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\, \] where $TV (g)$ denotes the classical total variation of a function of one real variable.

Cesari in [Ce] proposed the following modification of the Tonelli's plane variation, which is sometimes called Tonelli-Cesari variation

Definition Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli-Cesari variation $f$ as \[ V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\lambda\text{-a.e.}\right\}\, . \]

It turns out that the function $f$ has bounded Tonelli-Cesari variation if and only if it has bounded variation in the modern sense (see Variation of a function and Function of bounded variation). Among the many generalizations of the variation of a function of one variable given in the first half of the twentieth century, the Tonelli-Cesari is therefore the only one equivalent to the modern point of view. For a thorough discussion of the historical aspects of the theory of functions of bounded variation we refer to Section 3.12 of [AFP].

References

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Ce] L. Cesari, "Sulle funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 5 (1936) pp. 299-313.
[Co] D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Gi] E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
[To] L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 5 (1936) pp. 315-320.
How to Cite This Entry:
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=30111
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article