View source for Tonelli plane variation

{{MSC|26B30|26A45}} 

[[Category:Analysis]]

{{TEX|done}}

A generalization to functions of two variables of the [[Variation of a function]] of one variable, proposed by Tonelli in {{Cite|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  {{Cite|Ce}} proposed the following modification of the [[Tonelli plane variation|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 {{Cite|AFP}}. 

====References====
{|
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|valign="top"|{{Ref|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.      {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|Ce}}|| L. Cesari,  "Sulle  funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl.  Sci.  (2), '''5''' (1936) pp. 299-313.
|-
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
|-
|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of    functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,     1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Gi}}|| E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
|-
|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}
|-
|valign="top"|{{Ref|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. {{ZBL|0014.29606}}
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