Caccioppoli set

In mathematics, a Caccioppoli set is a set whose boundary is measurable and has a "finite measure". A synonym is set of finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation.

History

The basic concept of a Caccioppoli set was firstly introduced by the Italian mathematician Renato Caccioppoli in the paper Harv|Caccioppoli|1927: considering a plane set or a surface defined on a open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, "provided this quantity was bounded". The "measure of the boundary of a set was defined as a functional", precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and its value can be approximated by the values it takes on an increasing net of subsets. Another clearly stated (and demonstrated) property of this functional was its "lower semi-continuity". In the paper Harv|Caccioppoli|1928, he precised the concepts by using a "triangular mesh" as a increasing net approximating the open domain, defining "positive and negative variations" whose sum is the total variation, i.e. the "area functional". His inspiring point of view, as he explicitly admitted, was those of Giuseppe Peano, as expressed by the Peano-Jordan Measure: "associate to every portion of a surface a oriented plane area in a similar way as a approximating chord is associated to a curve". Also, another theme found in this theory was the "extension of a functional" from a subspace to the whole ambient space: the use of theorems generalizing the Hahn-Banach theorem is frequently encountered in Caccioppoli research. However, the restricted meaning of total variation in the sense of Tonelli added much complication to the formal development of the theory, and the use of a parametric description of the sets restricted its scope. Lamberto Cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936: maybe this was one of the reasons that induced Caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk Harv|Caccioppoli|1953 at the IV UMI Congress in October 1951, followed by five notes published in the [http://www.lincei.it/pubblicazioni/rendicontiFMN/inizio_eng.html Rendiconti] of the Accademia Nazionale dei Lincei. Further study by Ennio de Giorgi shown the fundamental results about their properties.

See also

* Bounded variation
* Renato Caccioppoli
* Herbert Federer
* Geometric measure theory
* Ennio de Giorgi
* Perimeter
* Pfeffer integral
* Total variation

References

*Harvrefcol
Surname = Caccioppoli
Given = Renato
Title = Sulla quadratura delle superfici piane e curve (On the quadrature of curved and plane surfaces)
Journal = [http://www.lincei.it/pubblicazioni/rendicontiFMN/inizio_eng.html Rendiconti dell'Accademia Nazionale dei Lincei]
Volume = 6 (serie 6)
Year = 1927
Page = 142-146 (in Italian). The first paper containing the seminal concept of what a Caccioppoli set is.
*Harvrefcol
Surname = Caccioppoli
Given = Renato
Title = Sulle coppie di funzioni a variazione limitata (On the couples of functions of bounded variation)
Journal = [http://www.socnazsla.unina.it/italiano/frameini.htm Rendiconti dell'Accademia di Scienze Fisiche e Matematiche di Napoli]
Volume = 34 (serie 3)
Year = 1928
Page = 83-88 (in Italian). A paper where the concepts introduced in the preceding paper are precised and extended.
*Harvrefcol
Surname = Caccioppoli
Given = Renato
Chapter = Elementi di una teoria generale dell’integrazione $k$-dimensionale in uno spazio $n$-dimensionale (Elements of a general theory of $k$-dimensional integration in a $n$-dimensional space).
Title = Atti IV Congresso U.M.I., Taormina, October 1951, vol. 2,
Publisher = Edizioni Cremonese
Place = Roma
Page = 41-49
Year = 1953 (in Italian). The first paper detailing the theory of finite perimeter set in a fairly complete setting.
*Harvrefcol
Surname = Caccioppoli
Given = Renato
Title = Opere scelte (Selected Papers)
Publisher = [http://www.ed-cremonese.it/ Edizioni Cremonese]
Place = Roma
Year = 1963 (in Italian), volume 1 (ISBN 88-708-3505-7) and 2 (ISBN 88-708-3506-5). A selection from Caccioppoli's scientific works with a biography and a commentary of Mauro Picone.
*Harvrefcol
Surname = Miranda
Given = Mario
Title = [http://www.lincei.it/pubblicazioni/rendicontiFMN/rol/visabs.php?lang=en&type=mat&fileId=234 Caccioppoli sets]
Journal = [http://www.lincei.it/pubblicazioni/rendicontiFMN/inizio_eng.html Rendiconti Lincei - Matematica e Applicazioni]
Volume = 14 (serie 9)
Year = 2003 | Page = 173-177. A paper sketching the history of the theory of sets of finite perimeter, from the seminal paper of Renato Caccioppoli to main discoveries.

Bibliography

*Harvrefcol
Surname1 = De Giorgi
Given1 = Ennio
Surname2 = Colombini
Given2 = Ferruccio
Surname3 = Piccinini
Given3 = Livio
Title = Frontiere orientate di misura minima e questioni collegate (Oriented boundaries of minimal measure and related questions)
Publisher = [http://www.sns.it/en/edizioni/edizioni/ Edizioni della Normale] ,
Place = Pisa
Year=1972 (in Italian). An advanced text, oriented to the theory of minimal surfaces in the multi-dimensional setting, written by one of the leading contributors.
*Harvrefcol
Surname = Federer
Given = Herbert
Title = Geometric Measure Theory
Publisher = [http://www.springer.com Springer Verlag]
Place = Berlin-Heidelberg
Year = 1969,1996 ISBN 3-540-60656-4, particularly chapter 4, paragraph 4.5, sections 4.5.1 to 4.5.4 "Sets with locally finite perimeter". The absolute reference text in geometric measure theory.
* Harvrefcol
Surname = Giusti
Given = Enrico
Title = Minimal surfaces and functions of bounded variations
Publisher = [http://www.birkhauser.com Birkhäuser Verlag]
Place = Basel
Year = 1984 ISBN 0-8176-3153-4, particularly part I, chapter 1 "Functions of bounded variation and Caccioppoli sets". A good reference, strongly oriented to applications to minimal surface theory.
*Harvrefcol
Surname1 = Vol'pert
Given1 = Aizik Isaakovich
Surname2 = Hudjaev
Given2 = Sergei Ivanovich
Title = Analysis in classes of discontinuous functions and equations of mathematical physics
Publisher = Martinus Nijhoff Publishers
Place = Dordrecht
Year=1986 ISBN 90-247-3109-7, particularly part II, chapter 4 paragraph 2 "Sets with finite perimeter". One of the best books about "BV"-functions and their application to problems of mathematical physics, particularly chemical kinetics.
*Harvrefcol
Surname = Vol'pert
Given = Aizik Isaakovich
Title = [http://www.math.technion.ac.il/~volp/spaces_BV.pdf Spaces BV and quasi-linear equations]
Journal = [http://www.turpion.org/php/homes/pa.phtml?jrnid=sm Mathematics USSR-Sbornik]
Volume = 2
Issue = 2
Page = 225-267
Year = 1967
Access-date = January 23, 2007. A seminal paper where Caccioppoli sets and "BV" functions are deeply studied and applied to the theory of partial differential equations.

External links

* Toby Christopher O'Neil " [http://eom.springer.de/g/g130040.htm Geometric measure theory] ", Springer-Verlag Online Encyclopaedia of Mathematics.
* Victor Abramovich Zagaller " [http://eom.springer.de/P/p072130.htm Perimeter] ", Springer-Verlag Online Encyclopaedia of Mathematics.