The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper (Caccioppoli 1927): considering a plane set or a surface defined on an 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.
Another clearly stated (and demonstrated) property of this functional was its lower semi-continuity.
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.
Lamberto Cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936:[1] 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 (Caccioppoli 1953) at the IV UMI Congress in October 1951, followed by five notes published in the Rendiconti of the Accademia Nazionale dei Lincei.
These notes were sharply criticized by Laurence Chisholm Young in the Mathematical Reviews.
[2] 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.
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.
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".
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.
is defined to be the total variation of its characteristic function on that open set.
is a Caccioppoli set if and only if it has finite perimeter in every bounded open subset
Therefore, a Caccioppoli set has a characteristic function whose total variation is locally bounded.
From the theory of functions of bounded variation it is known that this implies the existence of a vector-valued Radon measure
such that As noted for the case of general functions of bounded variation, this vector measure
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).
there exist two naturally associated analytic quantities: the vector-valued Radon measure
There is an elementary lemma that guarantees that the support (in the sense of distributions) of
turns out to be too crude for Caccioppoli sets because its Hausdorff measure overcompensates for the perimeter
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
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.
is the orthogonal complement of the unit vector defined previously.
, so it is interpreted as an approximate inward pointing unit normal vector to the reduced boundary
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