Difference between revisions of "Tonelli plane variation"
From Encyclopedia of Mathematics
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| − | + | {{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==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |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}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 10:54, 16 March 2023
2020 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. Zbl 0014.29606 |
How to Cite This Entry:
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=16914
Tonelli plane variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tonelli_plane_variation&oldid=16914
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article