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.

[edit] History

The basic concept of a Caccioppoli set was firstly introduced by the Italian mathematician Renato Caccioppoli in the paper (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 (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 (Caccioppoli 1953) at the IV UMI Congress in October 1951, followed by five notes published in the Rendiconti of the Accademia Nazionale dei Lincei. Further study by Ennio de Giorgi shown the fundamental results about their properties.

[edit] See also

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

[edit] References

• Caccioppoli, Renato (1927). Sulla quadratura delle superfici piane e curve (On the quadrature of curved and plane surfaces). Rendiconti dell'Accademia Nazionale dei Lincei 6 (serie 6):142-146 (in Italian). The first paper containing the seminal concept of what a Caccioppoli set is.
• Caccioppoli, Renato (1928). Sulle coppie di funzioni a variazione limitata (On the couples of functions of bounded variation). Rendiconti dell'Accademia di Scienze Fisiche e Matematiche di Napoli 34 (serie 3):83-88 (in Italian). A paper where the concepts introduced in the preceding paper are precised and extended.
• Caccioppoli, Renato (1953). "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)." in Atti IV Congresso U.M.I., Taormina, October 1951, vol. 2,(p. 41-49). Roma: Edizioni Cremonese (in Italian). The first paper detailing the theory of finite perimeter set in a fairly complete setting.
• Caccioppoli, Renato (1963). Opere scelte (Selected Papers). Roma: Edizioni Cremonese (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.
• Miranda, Mario (2003). Caccioppoli sets. Rendiconti Lincei - Matematica e Applicazioni 14 (serie 9):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.

[edit] Bibliography

• De Giorgi, Ennio, Colombini, Ferruccio & Piccinini, Livio (1972). Frontiere orientate di misura minima e questioni collegate (Oriented boundaries of minimal measure and related questions). Pisa: Edizioni della Normale, (in Italian). An advanced text, oriented to the theory of minimal surfaces in the multi-dimensional setting, written by one of the leading contributors.
• Federer, Herbert (1969,1996). Geometric Measure Theory. Berlin-Heidelberg: Springer Verlag 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.
• Giusti, Enrico (1984). Minimal surfaces and functions of bounded variations. Basel: Birkhäuser Verlag 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.
• Vol'pert, Aizik Isaakovich & Hudjaev, Sergei Ivanovich (1986). Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht: Martinus Nijhoff Publishers 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.
• Vol'pert, Aizik Isaakovich (1967). Spaces BV and quasi-linear equations. Mathematics USSR-Sbornik 2.2:225-267 [January 23, 2007]. A seminal paper where Caccioppoli sets and BV functions are deeply studied and applied to the theory of partial differential equations.

[edit] External links

• Toby Christopher O'Neil "Geometric measure theory", Springer-Verlag Online Encyclopaedia of Mathematics.
• Victor Abramovich Zagaller "Perimeter", Springer-Verlag Online Encyclopaedia of Mathematics.