In the paper (
Caccioppoli 1928), he precised by using a triangular mesh as an 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: to associate to every portion of a surface an
oriented plane area in a similar way as an
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.
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.[3] The same year he published his first paper on the topic i.e. (
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 (
De Giorgi 1954), reviewed again by Laurence Chisholm Young in the Mathematical Reviews,[4] 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
perimeters 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 (
Federer & Fleming 1960), changing the approach to the theory. Basically they introduced two new kind of
currents, respectively
normal currents and
integral currents: in a subsequent series of papers and in his famous treatise,[5] Federer showed that Caccioppoli sets are normal
currents of dimension in -dimensional
euclidean spaces. However, even if the theory of Caccioppoli sets can be studied within the framework of theory of
currents, it is customary to study it through the "traditional" approach using
functions of bounded variation, as the various sections found in a lot of important
monographs in
mathematics and
mathematical physics testify.[6]
Formal definition
In what follows, the definition and properties of
functions of bounded variation in the -dimensional setting will be used.
Actually De Giorgi considered the case : 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 (
Giusti 1984). Now having defined what a perimeter is, De Giorgi gives the same definition 2 of what a set of
(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 then , with equality holding if and only if the
closure of is a compact subset of .
For any two Cacciopoli sets and , the relation holds, with equality holding if and only if , where is the
distance between sets in
euclidean space.
If the
Lebesgue measure of is , then : this implies that if the
symmetric difference of two sets has zero Lebesgue measure, the two sets have the same perimeter i.e. .
Notions of boundary
For any given Caccioppoli set there exist two naturally associated analytic quantities: the vector-valued
Radon measure and its
total variation measure. Given that
is the perimeter within any open set , one should expect that alone should somehow account for the perimeter of .
The topological boundary
It is natural to try to understand the relationship between the objects , , and the
topological boundary. There is an elementary lemma that guarantees that the
support (in the sense of
distributions) of , and therefore also , is always contained in :
Proof. To see this choose : then belongs to the
open set and this implies that it belongs to an
open neighborhood contained in the
interior of or in the interior of . Let . If where is the
closure of , then for and
Likewise, if then for so
With arbitrary it follows that is outside the support of .
The reduced boundary
The topological boundary turns out to be too crude for Caccioppoli sets because its
Hausdorff measure overcompensates for the perimeter defined above. Indeed, the Caccioppoli set
representing a square together with a line segment sticking out on the left has perimeter , i.e. the extraneous line segment is ignored, while its topological boundary
has one-dimensional Hausdorff measure .
The "correct" boundary should therefore be a subset of . We define:
Definition 4. The reduced boundary of a Caccioppoli set is denoted by and is defined to be equal to be the collection of points at which the limit:
exists and has length equal to one, i.e. .
One can remark that by the
Radon-Nikodym Theorem the reduced boundary is necessarily contained in the support of , which in turn is contained in the topological boundary as explained in the section above. That is:
The inclusions above are not necessarily equalities as the previous example shows. In that example, is the square with the segment sticking out, is the square, and is the square without its four corners.
De Giorgi's theorem
For convenience, in this section we treat only the case where , i.e. the set 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
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 is a Caccioppoli set. Then at each point of the reduced boundary there exists a multiplicity one
approximate tangent space of , i.e. a codimension-1 subspace of such that
for every continuous, compactly supported . In fact the subspace is the
orthogonal complement of the unit vector
defined previously. This unit vector also satisfies
locally in , so it is interpreted as an approximate inward pointing
unitnormal vector to the reduced boundary . Finally, is (n-1)-
rectifiable and the restriction of (n-1)-dimensional
Hausdorff measure to is , i.e.
for all Borel sets .
In other words, up to -measure zero the reduced boundary is the smallest set on which is supported.
Applications
A Gauss–Green formula
From the definition of the vector
Radon measure and from the properties of the perimeter, the following formula holds true:
This is one version of the
divergence theorem for
domains with non smooth
boundary. De Giorgi's theorem can be used to formulate the same identity in terms of the reduced boundary and the approximate inward pointing unit normal vector . Precisely, the following equality holds
Ambrosio, Luigi (2010), "La teoria dei perimetri di Caccioppoli–De Giorgi e i suoi più recenti sviluppi" [The De Giorgi-Caccioppoli theory of perimeters and its most recent developments], Rendiconti Lincei - Matematica e Applicazioni, 9, 21 (3): 275–286,
doi:10.4171/RLM/572,
MR2677605,
Zbl1195.49052. 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.
Caccioppoli, Renato (1928), "Sulle coppie di funzioni a variazione limitata" [On pairs of functions of bounded variation], Rendiconti dell'Accademia di Scienze Fisiche e Matematiche di Napoli, 3 (in Italian), 34: 83–88,
JFM54.0290.04. The work where Caccioppoli made rigorous and developed the concepts introduced in the preceding paper (
Caccioppoli 1927).
Caccioppoli, Renato (1953), "Elementi di una teoria generale dell'integrazione k-dimensionale in uno spazio n-dimensionale", Atti IV Congresso U.M.I., Taormina, October 1951 [Elements of a general theory of k-dimensional integration in a n-dimensional space] (in Italian), vol. 2,
Roma: Edizioni Cremonese (distributed by
Unione Matematica Italiana), pp. 41–49,
MR0056067,
Zbl0051.29402.The first paper detailing the theory of finite perimeter set in a fairly complete setting.
De Giorgi, Ennio (1954), "Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni" [On a general theory of (r-1)-dimensional measure in r-dimensional space], Annali di Matematica Pura ed Applicata, Serie IV (in Italian), 36 (1): 191–213,
doi:
10.1007/BF02412838,
hdl:10338.dmlcz/126043,
MR0062214,
S2CID122418733,
Zbl0055.28504. The first complete exposition by De Giorgi of the theory of Caccioppoli sets.
De Giorgi, Ennio; Colombini, Ferruccio; Piccinini, Livio (1972), Frontiere orientate di misura minima e questioni collegate [Oriented boundaries of minimal measure and related questions], Quaderni (in Italian),
Pisa: Edizioni della Normale, p. 180,
MR0493669,
Zbl0296.49031. An advanced text, oriented towards the theory of
minimal surfaces in the multi-dimensional setting, written by one of the leading contributors.
Simon, Leon (1983), Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, vol. 3, Australian National University, particularly Chapter 3, Section 14 "Sets of Locally Finite Perimeter".