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Caccioppoli set

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

Contents

[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 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: 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. These notes were sharply criticized by Laurence Chisholm Young in the Mathematical Reviews. In 1952 Ennio de Giorgi presented to the Salzburg Congress of the Austrian Mathematical Society his first results related to the ideas of Caccioppoli about the definition of the measure of boundaries of sets: he obtained his results by using a smoothing operator analogous to a mollifier, constructed from the Gaussian function, proving some results of Caccioppoli in a different way. Probably he was led to study this theory by his teacher and friend Mauro Picone, who was also a friend of Caccioppoli, which he met in 1953 for the first time: Caccioppoli expressed a profund appreciation of his work, starting a friendship with De Giorgi. 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 by Laurence Chisholm Young in the Mathematical Reviews, 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 the famous treatise (Federer 1969,1996) Federer showed that Caccioppoli sets are normal currents of dimension n in n-dimensional euclidean spaces. However, even if the theory of Caccioppoli sets can be studied within the 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 (see the "Bibliography" section).

[edit] Formal definition

In what follows, the definition and properties of functions of bounded variation in the n-dimensional setting will be used.

[edit] Caccioppoli definition

Definition 1. Let Ω be an open subset of \scriptstyle\mathbb{R}^n and let E be a Borel set. The perimeter of E in Ω is defined as follows

P(E,\Omega) = \int_\Omega\vert D\chi_E\vert = V\left(\chi_E,\Omega\right):=\sup\left\{\int_E \mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x\colon \boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n),\ \Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\le 1\right\}

where χE is the characteristic function of E. If \scriptstyle\Omega = \mathbb{R}^n, then \scriptstyle P(E) = P(E,\mathbb{R}^n): as one can see from this definition, the perimeter of a general Borel set is the total variation of its characteristic function.

Definition 2. The Borel set E is a Caccioppoli set if and only if for every bounded open subset Ω of \scriptstyle\mathbb{R}^n it has a locally finite perimeter, i.e.

P(E,\Omega)<+\infty

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 Radon measure DχE such that

\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,\mathbb{R}^n)

As noted for the case of general functions of bounded variation, this vector measure DχE is the distributional or weak gradient of χE.

[edit] De Giorgi definition

In his papers (De Giorgi 1953) and (De Giorgi 1954), Ennio de Giorgi introduced the following smoothing operator, analogous to the Weierstrass transform in the one-dimensional case

W_\lambda\chi_E(x)=\int_{\mathbb{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

As one can esily prove, wλχ(x) is a smooth function for all \scriptstyle x\in\mathbb{R}^n, such that

\lim_{\lambda\to 0}W_\lambda\chi_E(x)=\chi_E(x)

also, its gradient is everywere well defined, and so is its absolute value

\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\vert DW_\lambda\chi_E(x)\right\vert = \sqrt{\sum_{k=1}^n\left|\frac{\partial W_\lambda\chi_E(x)}{\partial x_k}\right|^2}

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

Definition 3. et Ω be an open subset of \scriptstyle\mathbb{R}^n and let E be a Borel set. The perimeter of E in Ω is the value

P(E,\Omega) = \lim_{\lambda\to 0}\int_\Omega \vert DW_\lambda\chi_E(x)\vert\mathrm{d}x

Actually De Giorgi considered the case \scriptstyle\Omega=\mathbb{R}^n: 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.

[edit] Properties

[edit] Basic properties

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

  1. If \scriptstyle\Omega\subseteq\Omega_1 then \scriptstyle P(E\Omega)\leq P(E,\Omega_1), with equality holding if and only if the closure of E is a compact subset of Ω1.
  2. \scriptstyle P(E_1\cup E_2,\Omega)\leq P(E_1,\Omega) + P(E_2,\Omega_1) with equality holding if and only if d(E1,E2) > 0, where \scriptstyle d:\mathcal{P}(\mathbb{R}^n)\times\mathcal{P}(\mathbb{R}^n)\rightarrow\mathbb{R}^+_0 is the distance between sets in euclidean space.
  3. If the Lebesgue measure of E is 0, then P(E) = 0: this implies that if the symmetric difference \scriptstyle E_1\triangle E_2 of two sets has zero Lebesgue measure, the two sets have the same perimeter i.e. P(E1) = P(E2).

[edit] Support of the perimeter

Lemma 1. The support (in the sense of distributions) of the (vector) Radon measure DχE is a subset of the boundary of E, \scriptstyle\partial E. To see this choose \scriptstyle x\notin\partial E: then x belongs to the open set \scriptstyle\mathbb{R}^n\setminus\partial E and this implies that it belongs to an open neighborhood A contained in the interior of E or in the interior of \scriptstyle\mathbb{R}^n\setminus E. If \scriptstyle A\subseteq(\mathbb{R}^n\setminus E)^\circ=\mathbb{R}^n\setminus E^- where E − is the closure of E, then χE(x) = 0 on A and

\int_\Omega \langle\boldsymbol{\phi}, D\chi_E(x)\rangle = - \int_A\chi_E(x)\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = - \int_A0\cdot\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x =0 \qquad \forall\boldsymbol{\phi}\in C_c^1(A,\mathbb{R}^n)

thus x does not belong to the support suppDχE. Otherwise, if \scriptstyle A\subseteq E^\circ then χE(x) = 1 on A so

\int_\Omega \langle\boldsymbol{\phi}, D\chi_E(x)\rangle = - \int_A\mathrm{div}\boldsymbol{\phi}(x)\mathrm{d}x = 0 \qquad \forall\boldsymbol{\phi}\in C_c^1(A,\mathbb{R}^n)

[edit] Applications

[edit] A Gauss-Green formula

From the definition of the vector Radon measure DχE and from the properties of the perimeter, the following formula holds true:

\int_E\mathrm{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,\mathbb{R}^n)

As it is easily seen, this is a version of the divergence theorem for domains with non smooth boundary.

[edit] See also

[edit] References

[edit] Bibliography

[edit] External links

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