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In [[mathematics]], a '''Caccioppoli set''' is a [[Set (mathematics)|set]] whose [[boundary (topology)|boundary]] is [[measurable set|measurable]] and has (at least [[Local property|locally]]) a ''finite [[measure (mathematics)|measure]]''. A synonym is '''set of (locally) finite perimeter'''. Basically, a set is a Caccioppoli set if its [[Indicator function|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 (topology)|surface]] defined on an [[open set]] in the [[Plane (mathematics)|plane]], he defined their [[measure (mathematics)|measure]] or [[Area (geometry)|area]] as the [[total variation]] in the sense of [[Leonida Tonelli|Tonelli]] of their defining [[function (mathematics)|functions]], i.e. of their [[Surface (topology)#Extrinsically defined surfaces and embeddings|parametric equation]]s, ''provided this quantity was [[bounded variation|bounded]]''. The ''measure of the [[boundary (topology)|boundary of a set]] was defined as a '''[[functional (mathematics)|functional]]''''', precisely a [[set function]], for the first time: also, being defined on [[open set]]s, it can be defined on all [[Borel set]]s and its value can be approximated by the values it takes  on an increasing [[Net (mathematics)|net]] of [[subset]]s. Another clearly stated (and demonstrated) property of this functional was its ''[[semi-continuity|lower semi-continuity]]''.

In the paper {{Harv|Caccioppoli|1928}}, he precised  by using a ''[[Polygon mesh|triangular mesh]]'' as an increasing [[Net (mathematics)|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 [[Jordan measure|Peano-Jordan Measure]]: ''to associate to every portion of a surface an [[Orientation (mathematics)|oriented]] plane area in a similar way as an [[Chord (geometry)|approximating chord]] is associated to a curve''. Also, another theme found in this theory was the ''extension of a [[functional (mathematics)|functional]]'' from a [[Linear subspace|subspace]] to the whole [[linear space|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 [[Leonida Tonelli|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 [[Bounded variation|functions of bounded variation]] to the case of several variables only in 1936:<ref>In the paper {{harv|Cesari|1936}}. See the entries "[[Bounded variation]]" and "[[Total variation]]" for more details.</ref> perhaps, 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 [[Italian Mathematical Union|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]]. These notes were sharply criticized by [[Laurence Chisholm Young]] in the [[Mathematical Reviews]].<ref>See {{MR|56067}}.</ref>

In 1952 [[Ennio de Giorgi]] presented his first results, developing the ideas of Caccioppoli, on the definition of the measure of boundaries of sets at the [[Salzburg]] Congress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to a [[mollifier]], constructed from the [[Gaussian function]], independently proving some results of Caccioppoli. Probably he was led to study this theory by his teacher and friend [[Mauro Picone]], who had also been the teacher of Caccioppoli and was likewise his friend. De Giorgi met Caccioppoli in 1953 for the first time: during their meeting, Caccioppoli expressed a profound appreciation of his work, starting their lifelong friendship.<ref>It lasted up to the tragic death of Caccioppoli in 1959.</ref> The same year he published his first paper on the topic i.e. {{Harv|De Giorgi|1953}}: however, this paper and the closely following one did not attracted much interest from the mathematical community. It was only with the paper {{Harv|De Giorgi|1954}}, reviewed again by Laurence Chisholm Young in the Mathematical Reviews,<ref>See {{MR|0062214}}.</ref> that his approach to sets of finite perimeter became widely known and appreciated: also, in the review, Young revised  his previous criticism on the work of Caccioppoli.

The last paper of De Giorgi on the theory of [[perimeter]]s was published in 1958: in 1959, after the death of Caccioppoli, he started to call sets of finite perimeter "Caccioppoli sets". Two years later [[Herbert Federer]] and [[Wendell Fleming]] published their paper {{Harv|Federer|Fleming|1960}}, changing the approach to the theory. Basically they introduced two new kind of [[current (mathematics)|currents]], respectively [[normal current]]s and [[integral current]]s: in a subsequent series of papers and in his famous treatise,<ref>See {{Harv|Federer|1969}}.</ref> Federer showed that Caccioppoli sets are normal [[current (mathematics)|currents]] of dimension <math>n</math> in <math>n</math>-dimensional [[euclidean space]]s. However, even if the theory of Caccioppoli sets can be studied within the framework of theory of [[current (mathematics)|currents]], it is customary to study it through the "traditional" approach using [[Bounded variation|functions of bounded variation]], as the various sections found in a lot of important [[monograph]]s in [[mathematics]] and [[mathematical physics]] testify.<ref>See the "[[Caccioppoli set#References|References]]" section.</ref>

== Formal definition ==
In what follows, the definition and properties of [[Bounded variation|functions of bounded variation]] in the <math>n</math>-dimensional setting will be used.

=== Caccioppoli definition ===
'''Definition 1'''. Let ''<math>\Omega</math>'' be an [[open subset]] of <math>\R^n </math> and let <math>E</math> be a [[Borel set]]. The ''[[perimeter]] of <math>E</math> in <math>\Omega</math>'' is defined as follows

:<math>P(E,\Omega) = V\left(\chi_E,\Omega\right):=\sup\left\{\int_\Omega \chi_E(x) \mathrm{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : \boldsymbol{\phi}\in C_c^1(\Omega,\R^n),\ \|\boldsymbol{\phi}\|_{L^\infty(\Omega)}\le 1\right\}
</math>

where <math>\chi_E</math> is the [[Indicator function|characteristic function]] of <math>E</math>. That is, the perimeter of <math>E</math> in an open set <math>\Omega</math> is defined to be the [[total variation]] of its [[Indicator function|characteristic function]] on that open set. If <math>\Omega = \R^n</math>, then we write <math>P(E) = P(E,\R^n)</math> for the (global) perimeter.

'''Definition 2'''. The [[Borel set]] <math>E</math> is a '''Caccioppoli set''' if and only if it has finite perimeter in every [[bounded set|bounded]] [[open subset]] <math>\Omega</math> of <math>\R ^n </math>, i.e.

:<math>P(E,\Omega)<+\infty</math> whenever <math>\Omega \subset \R^n</math> is open and bounded.

Therefore, a Caccioppoli set has a [[Indicator function|characteristic function]] whose [[total variation]] is locally bounded. From the theory of [[Bounded variation|functions of bounded variation]] it is known that this implies the existence of a [[Euclidean vector|vector-valued]] [[Radon measure]] <math>D\chi_E</math> such that

:<math>\int_\Omega\chi_E(x)\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = \int_E\mathrm{div}\boldsymbol{\phi}(x) \, \mathrm{d}x = -\int_\Omega \langle\boldsymbol{\phi}, D\chi_E(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\R ^n)</math>

As noted for the case of general [[Bounded variation|functions of bounded variation]], this vector [[measure (mathematics)|measure]] <math>D\chi_E</math> is the [[Distribution (mathematics)#Formal definition|distributional]] or [[weak derivative|weak]] [[gradient]] of <math>\chi_E</math>. The total variation measure associated with <math>D\chi_E</math> is denoted by <math>|D\chi_E|</math>, i.e. for every open set <math>\Omega \subset \R^n</math> we write <math>|D\chi_E|(\Omega)</math> for <math>P(E, \Omega) = V(\chi_E, \Omega)</math>.

=== De Giorgi definition ===
In his papers {{Harv|De Giorgi|1953}} and {{Harv|De Giorgi|1954}}, [[Ennio de Giorgi]] introduces the following [[smoothing operator]], analogous to the [[Weierstrass transform]] in the one-[[Dimension (mathematics)|dimensional]] case

:<math>W_\lambda\chi_E(x)=\int_{\R ^n}g_\lambda(x-y)\chi_E(y)\mathrm{d}y = (\pi\lambda)^{-\frac{n}{2}}\int_Ee^{-\frac{(x-y)^2}{\lambda}}\mathrm{d}y</math>

As one can easily prove, <math>W_\lambda\chi(x)</math> is a [[smooth function]] for all <math>x\in\R^n</math>, such that

:<math>\lim_{\lambda\to 0}W_\lambda\chi_E(x)=\chi_E(x)</math>

also, its [[gradient]] is everywhere well defined, and so is its [[absolute value]]

:<math>\nabla W_\lambda\chi_E(x) = \mathrm{grad}W_\lambda\chi_E(x) = DW_\lambda\chi_E(x) = 
\begin{pmatrix}\frac{\partial W_\lambda\chi_E(x)}{\partial x_1}\\ \vdots\\ \frac{\partial W_\lambda\chi_E(x)}{\partial x_n}\\ \end{pmatrix} 
\Longleftrightarrow
\left | DW_\lambda\chi_E(x)\right | = \sqrt{\sum_{k=1}^n\left|\frac{\partial W_\lambda\chi_E(x)}{\partial x_k}\right|^2}</math>

Having defined this function, De Giorgi gives the following definition of [[perimeter]]:

'''Definition 3'''. Let '''<math> \Omega </math>''' be an [[open subset]] of <math>\R^n</math> and let <math>E</math> be a [[Borel set]]. The ''[[perimeter]] of <math>E</math> in <math>\Omega</math>'' is the value

:<math>P(E,\Omega) = \lim_{\lambda\to 0}\int_\Omega | DW_\lambda\chi_E(x) | \mathrm{d}x</math>

Actually De Giorgi considered the case <math>\Omega=\R ^n</math>: however, the extension to the general case is not difficult. It can be proved that the two definitions are exactly equivalent: for a proof see the already cited De Giorgi's papers or the book {{Harv|Giusti|1984}}. Now having defined what a perimeter is, De Giorgi gives the same definition 2 of what a set of [[Local property|(locally) finite]] perimeter is.

== Basic properties ==

The following properties are the ordinary properties which the general notion of a [[perimeter]] is supposed to have:

* If <math>\Omega\subseteq\Omega_1</math> then <math>P(E,\Omega)\leq P(E,\Omega_1)</math>, with equality holding if and only if the [[Closure (topology)|closure]] of <math>E</math> is a compact subset of <math>\Omega</math>.
* For any two Cacciopoli sets <math>E_1</math> and <math>E_2</math>, the relation <math>P(E_1\cup E_2,\Omega)\leq P(E_1,\Omega) + P(E_2,\Omega_1)</math> holds, with equality holding if and only if <math>d(E_1,E_2)>0</math>, where <math>d</math> is the [[Distance#Distances between sets and between a point and a set|distance between sets]] in [[euclidean space]].
* If the [[Lebesgue measure]] of <math>E</math> is <math>0</math>, then <math>P(E)=0</math>: this implies that if the [[symmetric difference]] <math>E_1\triangle E_2</math> of two sets has zero Lebesgue measure, the two sets have the same perimeter i.e. <math>P(E_1)=P(E_2)</math>.

== Notions of boundary ==

For any given Caccioppoli set <math>E \subset \R ^n</math> there exist two naturally associated analytic quantities: the vector-valued [[Radon measure]] <math>D\chi_E</math> and its [[Total variation#Total variation in measure theory|total variation measure]] <math>|D\chi_E|</math>. Given that

:<math> P(E, \Omega) = \int_{\Omega} |D\chi_E| </math>

is the perimeter within any open set <math>\Omega</math>, one should expect that <math>D\chi_E</math> alone should somehow account for the perimeter of <math>E</math>.

=== The topological boundary ===

It is natural to try to understand the relationship between the objects <math>D\chi_E</math>, <math>|D\chi_E|</math>, and the [[Boundary (topology)|topological boundary]] <math>\partial E</math>. There is an elementary lemma that guarantees that the [[Distribution (mathematics)#Support of a distribution|support]] (in the sense of [[Distribution (mathematics)|distributions]]) of <math>D\chi_E</math>, and therefore also <math>|D\chi_E|</math>, is always '''contained''' in <math>\partial E</math>:

'''Lemma'''. The support of the vector-valued Radon measure <math>D\chi_E</math> is a [[subset]] of the [[Boundary (topology)|topological boundary]] <math>\partial E</math> of <math>E</math>.

'''Proof'''. To see this choose <math>x_0 \notin\partial E</math>: then <math>x_0</math> belongs to the [[open set]] <math>\R ^n\setminus\partial E</math> and this implies that it belongs to an [[open neighborhood]] <math>A</math> contained in the [[Interior (topology)|interior]] of <math>E</math> or in the interior of <math>\R^n\setminus E</math>. Let <math>\phi \in C^1_c(A; \R ^n)</math>. If <math>A\subseteq(\R^n \setminus E)^\circ=\R^n\setminus E^-</math> where <math>E^-</math> is the [[Closure (topology)|closure]] of <math>E</math>, then <math>\chi_E(x)=0</math> for <math>x \in A</math> and

:<math> \int_\Omega \langle\boldsymbol{\phi}, D\chi_E(x)\rangle =- \int_A\chi_E(x) \, \operatorname{div}\boldsymbol{\phi}(x)\, \mathrm{d}x = 0</math>

Likewise, if <math>A\subseteq E^\circ </math> then <math>\chi_E(x)=1</math> for <math>x \in A</math> so

:<math>\int_\Omega \langle\boldsymbol{\phi}, D\chi_E(x)\rangle = -\int_A\operatorname{div} \boldsymbol{\phi}(x) \, \mathrm{d}x = 0
</math>

With <math>\phi \in C^1_c(A, \R^n)</math> arbitrary it follows that <math>x_0</math> is outside the support of <math>D\chi_E</math>.

=== The reduced boundary ===

The topological boundary <math>\partial E</math> turns out to be too crude for Caccioppoli sets because its [[Hausdorff measure]] overcompensates for the perimeter <math>P(E)</math> defined above. Indeed, the Caccioppoli set

:<math>E = \{ (x,y) : 0 \leq x, y \leq 1 \} \cup \{ (x, 0) : -1 \leq x \leq 1 \} \subset \R^2 </math>

representing a square together with a line segment sticking out on the left has perimeter <math>P(E) = 4</math>, i.e. the extraneous line segment is ignored, while its topological boundary

:<math>\partial E = \{(x, 0) : -1 \leq x \leq 1 \} \cup \{(x, 1) : 0 \leq x \leq 1 \} \cup \{(x, y) : x \in \{0, 1\}, 0 \leq y \leq 1 \} </math>

has one-dimensional Hausdorff measure <math>\mathcal{H}^1(\partial E) = 5</math>.

The "correct" boundary should therefore be a subset of <math>\partial E</math>. We define:

'''Definition 4'''. The '''reduced boundary''' of a Caccioppoli set <math>E \subset \R ^n</math> is denoted by <math>\partial^* E</math> and is defined to be equal to be the collection of points <math>x</math> at which the limit:

:<math> \nu_E(x) := \lim_{\rho \downarrow 0} \frac{D\chi_E(B_\rho(x))}{|D\chi_E|(B_\rho(x))} \in \R^n</math>

exists and has length equal to one, i.e. <math>|\nu_E(x)| = 1</math>.

One can remark that by the [[Radon-Nikodym Theorem]] the reduced boundary <math>\partial^* E</math> is necessarily contained in the support of <math>D\chi_E</math>, which in turn is contained in the topological boundary <math>\partial E</math> as explained in the section above. That is:

:<math>\partial^* E \subseteq \operatorname{support} D\chi_E \subseteq \partial E</math>

The inclusions above are not necessarily equalities as the previous example shows. In that example, <math>\partial E</math> is the square with the segment sticking out, <math>\operatorname{support} D\chi_E</math> is the square, and <math>\partial^* E</math> is the square without its four corners.

=== De Giorgi's theorem ===

For convenience, in this section we treat only the case where <math>\Omega = \R ^n</math>, i.e. the set <math>E</math> has (globally) finite perimeter. De Giorgi's theorem provides geometric intuition for the notion of reduced boundaries and confirms that it is the more natural definition for Caccioppoli sets by showing

:<math> P(E) \left( = \int |D\chi_E| \right) = \mathcal{H}^{n-1}(\partial^* E)</math>

i.e. that its [[Hausdorff measure]] equals the perimeter of the set. The statement of the theorem is quite long because it interrelates various geometric notions in one fell swoop.

'''Theorem'''. Suppose <math>E \subset \R^n</math> is a Caccioppoli set. Then at each point <math>x</math> of the reduced boundary <math>\partial^* E</math> there exists a multiplicity one [[approximate tangent space]] <math>T_x</math> of <math>|D\chi_E|</math>, i.e. a codimension-1 subspace <math>T_x</math> of <math>\R ^n</math> such that

:<math> \lim_{\lambda \downarrow 0} \int_{\R^n} f(\lambda^{-1}(z-x)) |D\chi_E|(z) = \int_{T_x} f(y) \, d\mathcal{H}^{n-1}(y)</math>

for every continuous, compactly supported <math>f : \R ^n \to \R </math>. In fact the subspace <math>T_x</math> is the [[orthogonal complement]] of the unit vector

:<math>\nu_E(x) = \lim_{\rho \downarrow 0} \frac{D\chi_E(B_\rho(x))}{|D\chi_E|(B_\rho(x))} \in \R ^n</math>

defined previously. This unit vector also satisfies

:<math>\lim_{\lambda \downarrow 0} \left \{ \lambda^{-1}(z - x) : z \in E \right \} \to \left \{ y \in \R^n : y \cdot \nu_E(x) > 0 \right \}</math>

locally in <math>L^1</math>, so it is interpreted as an approximate inward pointing [[Unit vector|unit]] [[Normal (geometry)|normal vector]] to the reduced boundary <math>\partial^* E</math>. Finally, <math>\partial^* E</math> is (n-1)-[[Rectifiable set|rectifiable]] and the restriction of (n-1)-dimensional [[Hausdorff measure]] <math>\mathcal{H}^{n-1}</math> to <math>\partial^* E</math> is <math>|D\chi_E|</math>, i.e.

:<math>|D\chi_E|(A) = \mathcal{H}^{n-1}(A \cap \partial^* E)</math> for all Borel sets <math>A \subset \R^n</math>.

In other words, up to <math>\mathcal{H}^{n-1}</math>-measure zero the reduced boundary <math>\partial^* E</math> is the smallest set on which <math>D\chi_E</math> is supported.

== Applications ==

=== A Gauss–Green formula ===
From the definition of the vector [[Radon measure]] <math>D\chi_E</math> and from the properties of the perimeter, the following formula holds true:

:<math>\int_E\operatorname{div}\boldsymbol{\phi}(x) \, \mathrm{d}x = -\int_{\partial E} \langle\boldsymbol{\phi}, D\chi_E(x)\rangle 
\qquad \boldsymbol{\phi}\in C_c^1(\Omega, \R^n)</math>

This is one version of the [[divergence theorem]] for [[Domain (mathematics)#Real and complex analysis|domains]] with non smooth [[boundary (topology)|boundary]]. De Giorgi's theorem can be used to formulate the same identity in terms of the reduced boundary <math>\partial^* E</math> and the approximate inward pointing unit normal vector <math>\nu_E</math>. Precisely, the following equality holds

:<math>\int_E \operatorname{div} \boldsymbol{\phi}(x) \, \mathrm{d}x = - \int_{\partial^* E} \boldsymbol{\phi}(x) \cdot \nu_E(x) \, \mathrm{d}\mathcal{H}^{n-1}(x) \qquad \boldsymbol{\phi} \in C^1_c(\Omega, \R^n)</math>

== See also ==
{{div col}}
* [[Geometric measure theory]]
* [[Divergence theorem]]
* [[Pfeffer integral]]
{{div col end}}

== Notes ==
{{reflist|29em}}

==References==
{{refbegin}}

===Historical references===
*{{Citation
|last = Ambrosio
|first = Luigi
|author-link = Luigi Ambrosio
|title = La teoria dei perimetri di Caccioppoli–De Giorgi e i suoi più recenti sviluppi
|trans-title=The De Giorgi-Caccioppoli theory of perimeters and its most recent developments
|journal = [[Rendiconti Lincei - Matematica e Applicazioni]]
|volume = 21
|series = 9
|issue = 3
|pages = 275–286
|date = 
|year = 2010
|doi = 10.4171/RLM/572
|mr = 2677605 
|zbl = 1195.49052

}}. A paper surveying the history of the theory of sets of finite perimeter, from the seminal paper of [[Renato Caccioppoli]] and the contributions of [[Ennio De Giorgi]] to some more recent developments and open problems in metric measure spaces, in Carnot groups and in infinite-dimensional Gaussian spaces.
*{{Citation
|last = Caccioppoli
|first = Renato
|author-link = Renato Caccioppoli
|title = Sulla quadratura delle superfici piane e curve
|trans-title=On the quadrature of plane and curved surfaces
|journal = [[Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali]]
|series = VI
|language = Italian
|volume = 6
|issue = 
|pages = 142–146
|year = 1927
|jfm = 53.0214.02
}}. The first paper containing the seminal concept of what a Caccioppoli set is.
*{{Citation
|last = Caccioppoli
|first = Renato
|author-link = Renato Caccioppoli
|title = Sulle coppie di funzioni a variazione limitata
|trans-title=On pairs of functions of bounded variation
|journal = Rendiconti dell'Accademia di Scienze Fisiche e Matematiche di Napoli
|language = Italian
|volume = 34 
|series = 3
|pages = 83–88
|year = 1928
|jfm = 54.0290.04
}}. The work where Caccioppoli made rigorous and developed the concepts introduced in the preceding paper {{harv|Caccioppoli|1927}}.
*{{Citation
|last = Caccioppoli
|first = Renato
|author-link = Renato Caccioppoli
|contribution = Elementi di una teoria generale dell’integrazione {{mvar|k}}-dimensionale in uno spazio {{mvar|n}}-dimensionale 
|trans-title=Elements of a general theory of {{mvar|k}}-dimensional integration in a {{mvar|n}}-dimensional space
|language = Italian
|title = Atti IV Congresso U.M.I., [[Taormina]], October 1951
|year = 1953
|volume = 2
|pages = 41–49
|place = [[Rome|Roma]]
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|doi = 
|mr = 0056067 
|zbl = 0051.29402
}}.The first paper detailing the theory of finite perimeter set in a fairly complete setting.
*{{Citation
|last = Caccioppoli
|first = Renato
|author-link = Renato Caccioppoli
|title = Opere scelte
|trans-title=Selected papers
|place = [[Rome|Roma]]
|publisher = Edizioni Cremonese (distributed by [[Unione Matematica Italiana]])
|year = 1963
|pages = XXX+434 (vol. 1), 350 (vol. 2)
|doi = 
|mr = 
|zbl = 0112.28201
|isbn = 88-7083-505-7 
}}. A selection from Caccioppoli's scientific works with a biography and a commentary of [[Mauro Picone]].
*{{Citation
|last = Cesari
|first = Lamberto
|title = Sulle funzioni a variazione limitata
|trans-title=On the functions of bounded variation
|journal = [[Annali della Scuola Normale Superiore]]
|series = Serie II, 
|volume = 5
|issue = 3–4
|pages = 299–313
|date = 
|language = Italian
|year = 1936
|url = http://www.numdam.org/item?id=ASNSP_1936_2_5_3-4_299_0
|doi = 
|mr = 1556778
|zbl = 0014.29605
}}. Available at [http://www.numdam.org Numdam]. Cesari's watershed paper, where he extends the now called ''[[Total variation#Tonelli plane variation|Tonelli plane variation]]'' concept to include in the definition a subclass of the class of integrable functions.
*{{Citation
|last = De Giorgi
|first = Ennio
|author-link = 
|title = Definizione ed espressione analitica del perimetro di un insieme 
|trans-title=Definition and analytical expression of the perimeter of a set
|journal = [[Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali]]
|series = VIII
|language = Italian
|volume = 14
|pages = 390–393
|year = 1953
|doi = 
|mr = 0056066
|zbl =0051.29403
}}. The first note published by De Giorgi describing his approach to Caccioppoli sets.
*{{Citation
|last = De Giorgi
|first = Ennio
|author-link =  
|title = Su una teoria generale della misura {{math|(''r''-1)}}-dimensionale in uno spazio ad {{mvar|r}} dimensioni 
|trans-title=On a general theory of {{math|(''r''-1)}}-dimensional measure in {{mvar|r}}-dimensional space
|journal = Annali di Matematica Pura ed Applicata
|language = Italian
|series = Serie IV,
|volume = 36
|issue = 1
|pages = 191–213
|year = 1954
|doi = 10.1007/BF02412838
|mr = 0062214 
|zbl = 0055.28504
}}. The first complete exposition by De Giorgi of the theory of Caccioppoli sets.
*{{Citation
|last =  Federer
|first = Herbert
|author-link = Herbert Federer
|last2 = Fleming
|first2 = Wendell H.
|author2-link = 
|title = Normal and integral currents
|journal = [[Annals of Mathematics]]
|series = Series II,
|volume = 72
|issue = 4
|pages = 458–520
|year = 1960
|doi = 10.2307/1970227
|mr = 0123260 
|zbl = 0187.31301
|jstor =1970227
}}. The first paper of Herbert Federer illustrating his approach to the theory of perimeters based on the theory of currents.
*{{Citation
|last = Miranda
|first = Mario
|author-link = 
|title = Caccioppoli sets
|journal = [[Atti della Accademia Nazionale dei Lincei, Rendiconti Lincei, Matematica e Applicazioni]]
|series = IX 
|volume = 14
|issue = 3
|pages = 173–177
|year = 2003
|url = http://www.lincei.it/pubblicazioni/rendicontiFMN/rol/visabs.php?lang=en&type=mat&fileId=234 
|mr = 2064264 
|zbl = 1072.49030
}}. A paper sketching the history of the theory of sets of finite perimeter, from the seminal paper of [[Renato Caccioppoli]] to main discoveries.

===Scientific references===
*{{Citation
|last = De Giorgi
|first = Ennio
|author-link = Ennio de Giorgi
|last2 = Colombini
|first2 = Ferruccio
|author2-link = 
|last3 = Piccinini
|first3 = Livio
|title = Frontiere orientate di misura minima e questioni collegate
|trans-title=Oriented boundaries of minimal measure and related questions
|language = Italian
|place = [[Pisa]]
|publisher = Edizioni della Normale
|year = 1972
|series = Quaderni
|page = 180
|mr = 493669 
|zbl = 0296.49031
}}. An advanced text, oriented towards the theory of [[minimal surface]]s in the multi-dimensional setting, written by one of the leading contributors.
*{{citation
|last = Federer
|first = Herbert
|authorlink = Herbert Federer
|title = Geometric measure theory
|publisher = [[Springer-Verlag]] New York Inc.
|series = Classics in Mathematics
|volume = 
|location = [[Berlin]]-[[Heidelberg]]-[[New York City]]
|year = 1996
|origyear = 1969
|pages = xiv+676
|isbn = 3-540-60656-4
|mr= 0257325
|zbl = 0176.00801
}}, 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]].
*{{citation
|last=Simon
|first=Leon
|authorlink = Leon Simon
|title=Lectures on Geometric Measure Theory
|publisher=Australian National University
|series=Proceedings of the Centre for Mathematical Analysis
|year=1983
|volume=3
}}, particularly Chapter 3, Section 14 "''Sets of Locally Finite Perimeter''".
*{{Citation
|last = Giusti
|first = Enrico
|author-link = Enrico Giusti
|title = Minimal surfaces and functions of bounded variations
|place = [[Basel]]-[[Boston]]-[[Stuttgart]]
|publisher = [[Birkhäuser Verlag]]
|year = 1984 
|series = Monographs in Mathematics
|volume = 80
|pages = xii+240
|url = https://books.google.com/books?id=dNgsmArDoeQC&printsec=frontcover&dq=Minimal+surfaces+and+functions+of+bounded+variations
|mr = 0775682 
|zbl = 0545.49018
|isbn = 0-8176-3153-4}},  particularly part I, chapter 1 "''Functions of bounded variation and Caccioppoli sets''". A good reference on the theory of Caccioppoli sets and their application to the [[Minimal surface]] problem.
*{{Citation
|last = Hudjaev
|first = Sergei Ivanovich
|author-link = 
|last2 = Vol'pert
|first2 = Aizik Isaakovich
|author2-link = Aizik Isaakovich Vol'pert
|title = Analysis in classes of discontinuous functions and equations of mathematical physics
|place =  Dordrecht-Boston-Lancaster
|publisher = Martinus Nijhoff Publishers
|year = 1985
|series = Mechanics: analysis 
|volume = 8
|pages = xviii+678 
|url = https://books.google.com/books?id=lAN0b0-1LIYC&printsec=frontcover&dq=%22Analysis+in+classes+of+discontinuous+functions%22
|mr = 0785938
|zbl= 0564.46025
|isbn = 90-247-3109-7
}}, particularly part II, chapter 4 paragraph 2 "''Sets with finite perimeter''". One of the best books about {{mvar|BV}}–functions and their application to problems of [[mathematical physics]], particularly [[chemical kinetics]].
*{{Citation
|last = Maz'ya
|first = Vladimir G. 
|authorlink = Vladimir Gilelevich Maz'ya
|title = Sobolev Spaces
|publisher = [[Springer-Verlag]]
|location = [[Berlin]]–[[Heidelberg]]–-[[New York City]]
|year = 1985
|pages = xix+486
|isbn=3-540-13589-8
|mr = 817985 
|zbl = 0692.46023
}}; particularly chapter 6, "On functions in the space {{math|''BV''(Ω)}}". One of the best monographs on the theory of [[Sobolev spaces]].
*{{Citation
|last = Vol'pert
|first = Aizik Isaakovich
|author-link = Aizik Isaakovich Vol'pert
|title = Spaces {{mvar|BV}} and quasi-linear equations
|language = Russian
|journal = [[Matematicheskii Sbornik]]
|series = (N.S.),
|volume = 73(115)
|issue = 2
|pages = 255–302  
|year = 1967
|url = http://mi.mathnet.ru/eng/msb/v115/i2/p255
|mr = 216338
|zbl = 0168.07402
}}. A seminal paper where Caccioppoli sets and {{mvar|BV}}–functions are deeply studied and the concept of [[functional superposition]] is introduced and applied to the theory of [[partial differential equation]]s.
{{refend}}

== External links ==
*{{springer
 | title=Geometric measure theory
 | id= g/g130040
 | last= O'Neil
 | first=Toby Christopher
 | author-link=
}}
*{{springer
 | title=Perimeter
 | id= P/p072130
 | last=Zagaller
 | first=Victor Abramovich
 | author-link= Victor Abramovich Zalgaller
}}
*[http://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation Function of bounded variation] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

{{DEFAULTSORT:Caccioppoli Set}}
[[Category:Mathematical analysis]]
[[Category:Calculus of variations]]
[[Category:Measure theory]]