Tonelli plane variation
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. |
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=30111