Caccioppoli set#De Giorgi definition

The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper {{Harv|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. 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 {{Harv|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:In the paper {{harv|Cesari|1936}}. See the entries "Bounded variation" and "Total variation" for more details. 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 UMI Congress in October 1951, followed by five notes published in the [https://web.archive.org/web/20061227050646/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.See {{MR|56067}}.

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.It lasted up to the tragic death of Caccioppoli in 1959. 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,See {{MR|0062214}}. 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 {{Harv|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,See {{Harv|Federer|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 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.See the "References" section.

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

Definition 1. Let $\backslash Omega$ be an open subset of $\backslash R^n$ and let $E$ be a Borel set. The perimeter of $E$ in $\backslash Omega$ is defined as follows

:$P(E,\backslash Omega)\; =\; V\backslash left(\backslash chi\_E,\backslash Omega\backslash right):=\backslash sup\backslash left\backslash \{\backslash int\_\backslash Omega\; \backslash chi\_E(x)\; \backslash mathrm\{div\}\backslash boldsymbol\{\backslash phi\}(x)\; \backslash ,\; \backslash mathrm\{d\}x\; :\; \backslash boldsymbol\{\backslash phi\}\backslash in\; C\_c^1(\backslash Omega,\backslash R^n),\backslash \; \backslash |\backslash boldsymbol\{\backslash phi\}\backslash |\_\{L^\backslash infty(\backslash Omega)\}\backslash le\; 1\backslash right\backslash \}$

where $\backslash chi\_E$ is the characteristic function of $E$. That is, the perimeter of $E$ in an open set $\backslash Omega$ is defined to be the total variation of its characteristic function on that open set. If $\backslash Omega\; =\; \backslash R^n$, then we write $P(E)\; =\; P(E,\backslash R^n)$ for the (global) perimeter.

Definition 2. The Borel set $E$ is a Caccioppoli set if and only if it has finite perimeter in every bounded open subset $\backslash Omega$ of $\backslash R\; ^n$, i.e.

: whenever $\backslash Omega\; \backslash subset\; \backslash R^n$ is open and bounded.

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 $D\backslash chi\_E$ such that

:$\backslash int\_\backslash Omega\backslash chi\_E(x)\backslash mathrm\{div\}\backslash boldsymbol\{\backslash phi\}(x)\backslash mathrm\{d\}x\; =\; \backslash int\_E\backslash mathrm\{div\}\backslash boldsymbol\{\backslash phi\}(x)\; \backslash ,\; \backslash mathrm\{d\}x\; =\; -\backslash int\_\backslash Omega\; \backslash langle\backslash boldsymbol\{\backslash phi\},\; D\backslash chi\_E(x)\backslash rangle\; \backslash qquad\; \backslash forall\backslash boldsymbol\{\backslash phi\}\backslash in\; C\_c^1(\backslash Omega,\backslash R\; ^n)$

As noted for the case of general functions of bounded variation, this vector measure $D\backslash chi\_E$ is the distributional or weak gradient of $\backslash chi\_E$. The total variation measure associated with $D\backslash chi\_E$ is denoted by $|D\backslash chi\_E|$, i.e. for every open set $\backslash Omega\; \backslash subset\; \backslash R^n$ we write $|D\backslash chi\_E|(\backslash Omega)$ for $P(E,\; \backslash Omega)\; =\; V(\backslash chi\_E,\; \backslash Omega)$.

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-dimensional case

:$W\_\backslash lambda\backslash chi\_E(x)=\backslash int\_\{\backslash R\; ^n\}g\_\backslash lambda(x-y)\backslash chi\_E(y)\backslash mathrm\{d\}y\; =\; (\backslash pi\backslash lambda)^\{-\backslash frac\{n\}\{2\}\}\backslash int\_Ee^\{-\backslash frac\{(x-y)^2\}\{\backslash lambda\}\}\backslash mathrm\{d\}y$

As one can easily prove, $W\_\backslash lambda\backslash chi(x)$ is a smooth function for all $x\backslash in\backslash R^n$, such that

:$\backslash lim\_\{\backslash lambda\backslash to\; 0\}W\_\backslash lambda\backslash chi\_E(x)=\backslash chi\_E(x)$

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

:$\backslash nabla\; W\_\backslash lambda\backslash chi\_E(x)\; =\; \backslash mathrm\{grad\}W\_\backslash lambda\backslash chi\_E(x)\; =\; DW\_\backslash lambda\backslash chi\_E(x)\; =$

\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}

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

Definition 3. Let $\backslash Omega$ be an open subset of $\backslash R^n$ and let $E$ be a Borel set. The perimeter of $E$ in $\backslash Omega$ is the value

:$P(E,\backslash Omega)\; =\; \backslash lim\_\{\backslash lambda\backslash to\; 0\}\backslash int\_\backslash Omega\; |\; DW\_\backslash lambda\backslash chi\_E(x)\; |\; \backslash mathrm\{d\}x$

Actually De Giorgi considered the case $\backslash Omega=\backslash R\; ^n$: 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 {{Harv|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.

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

• If $\backslash Omega\backslash subseteq\backslash Omega\_1$ then $P(E,\backslash Omega)\backslash leq\; P(E,\backslash Omega\_1)$, with equality holding if and only if the closure of $E$ is a compact subset of $\backslash Omega$.
• For any two Cacciopoli sets $E\_1$ and $E\_2$, the relation $P(E\_1\backslash cup\; E\_2,\backslash Omega)\backslash leq\; P(E\_1,\backslash Omega)\; +\; P(E\_2,\backslash Omega\_1)$ holds, with equality holding if and only if $d(E\_1,E\_2)>0$, where $d$ is the distance between sets in euclidean space.
• If the Lebesgue measure of $E$ is $0$, then $P(E)=0$: this implies that if the symmetric difference $E\_1\backslash triangle\; E\_2$ of two sets has zero Lebesgue measure, the two sets have the same perimeter i.e. $P(E\_1)=P(E\_2)$.

For any given Caccioppoli set $E\; \backslash subset\; \backslash R\; ^n$ there exist two naturally associated analytic quantities: the vector-valued Radon measure $D\backslash chi\_E$ and its total variation measure $|D\backslash chi\_E|$. Given that

:$P(E,\; \backslash Omega)\; =\; \backslash int\_\{\backslash Omega\}\; |D\backslash chi\_E|$

is the perimeter within any open set $\backslash Omega$, one should expect that $D\backslash chi\_E$ alone should somehow account for the perimeter of $E$.

It is natural to try to understand the relationship between the objects $D\backslash chi\_E$, $|D\backslash chi\_E|$, and the topological boundary $\backslash partial\; E$. There is an elementary lemma that guarantees that the support (in the sense of distributions) of $D\backslash chi\_E$, and therefore also $|D\backslash chi\_E|$, is always contained in $\backslash partial\; E$:

Lemma. The support of the vector-valued Radon measure $D\backslash chi\_E$ is a subset of the topological boundary $\backslash partial\; E$ of $E$.

Proof. To see this choose $x\_0\; \backslash notin\backslash partial\; E$: then $x\_0$ belongs to the open set $\backslash R\; ^n\backslash setminus\backslash partial\; E$ and this implies that it belongs to an open neighborhood $A$ contained in the interior of $E$ or in the interior of $\backslash R^n\backslash setminus\; E$. Let $\backslash phi\; \backslash in\; C^1\_c(A;\; \backslash R\; ^n)$. If $A\backslash subseteq(\backslash R^n\; \backslash setminus\; E)^\backslash circ=\backslash R^n\backslash setminus\; E^-$ where $E^-$ is the closure of $E$, then $\backslash chi\_E(x)=0$ for $x\; \backslash in\; A$ and

:$\backslash int\_\backslash Omega\; \backslash langle\backslash boldsymbol\{\backslash phi\},\; D\backslash chi\_E(x)\backslash rangle\; =-\; \backslash int\_A\backslash chi\_E(x)\; \backslash ,\; \backslash operatorname\{div\}\backslash boldsymbol\{\backslash phi\}(x)\backslash ,\; \backslash mathrm\{d\}x\; =\; 0$

Likewise, if $A\backslash subseteq\; E^\backslash circ$ then $\backslash chi\_E(x)=1$ for $x\; \backslash in\; A$ so

:$\backslash int\_\backslash Omega\; \backslash langle\backslash boldsymbol\{\backslash phi\},\; D\backslash chi\_E(x)\backslash rangle\; =\; -\backslash int\_A\backslash operatorname\{div\}\; \backslash boldsymbol\{\backslash phi\}(x)\; \backslash ,\; \backslash mathrm\{d\}x\; =\; 0$

With $\backslash phi\; \backslash in\; C^1\_c(A,\; \backslash R^n)$ arbitrary it follows that $x\_0$ is outside the support of $D\backslash chi\_E$.

The topological boundary $\backslash partial\; E$ turns out to be too crude for Caccioppoli sets because its Hausdorff measure overcompensates for the perimeter $P(E)$ defined above. Indeed, the Caccioppoli set

:$E\; =\; \backslash \{\; (x,y)\; :\; 0\; \backslash leq\; x,\; y\; \backslash leq\; 1\; \backslash \}\; \backslash cup\; \backslash \{\; (x,\; 0)\; :\; -1\; \backslash leq\; x\; \backslash leq\; 1\; \backslash \}\; \backslash subset\; \backslash R^2$

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

:$\backslash partial\; E\; =\; \backslash \{(x,\; 0)\; :\; -1\; \backslash leq\; x\; \backslash leq\; 1\; \backslash \}\; \backslash cup\; \backslash \{(x,\; 1)\; :\; 0\; \backslash leq\; x\; \backslash leq\; 1\; \backslash \}\; \backslash cup\; \backslash \{(x,\; y)\; :\; x\; \backslash in\; \backslash \{0,\; 1\backslash \},\; 0\; \backslash leq\; y\; \backslash leq\; 1\; \backslash \}$

has one-dimensional Hausdorff measure $\backslash mathcal\{H\}^1(\backslash partial\; E)\; =\; 5$.

The "correct" boundary should therefore be a subset of $\backslash partial\; E$. We define:

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

:$\backslash nu\_E(x)\; :=\; \backslash lim\_\{\backslash rho\; \backslash downarrow\; 0\}\; \backslash frac\{D\backslash chi\_E(B\_\backslash rho(x))\}\{|D\backslash chi\_E|(B\_\backslash rho(x))\}\; \backslash in\; \backslash R^n$

exists and has length equal to one, i.e. $|\backslash nu\_E(x)|\; =\; 1$.

One can remark that by the Radon-Nikodym Theorem the reduced boundary $\backslash partial^*\; E$ is necessarily contained in the support of $D\backslash chi\_E$, which in turn is contained in the topological boundary $\backslash partial\; E$ as explained in the section above. That is:

:$\backslash partial^*\; E\; \backslash subseteq\; \backslash operatorname\{support\}\; D\backslash chi\_E\; \backslash subseteq\; \backslash partial\; E$

The inclusions above are not necessarily equalities as the previous example shows. In that example, $\backslash partial\; E$ is the square with the segment sticking out, $\backslash operatorname\{support\}\; D\backslash chi\_E$ is the square, and $\backslash partial^*\; E$ is the square without its four corners.

For convenience, in this section we treat only the case where $\backslash Omega\; =\; \backslash R\; ^n$, i.e. the set $E$ 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

:$P(E)\; \backslash left(\; =\; \backslash int\; |D\backslash chi\_E|\; \backslash right)\; =\; \backslash mathcal\{H\}^\{n-1\}(\backslash partial^*\; E)$

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 $E\; \backslash subset\; \backslash R^n$ is a Caccioppoli set. Then at each point $x$ of the reduced boundary $\backslash partial^*\; E$ there exists a multiplicity one approximate tangent space $T\_x$ of $|D\backslash chi\_E|$, i.e. a codimension-1 subspace $T\_x$ of $\backslash R\; ^n$ such that

:$\backslash lim\_\{\backslash lambda\; \backslash downarrow\; 0\}\; \backslash int\_\{\backslash R^n\}\; f(\backslash lambda^\{-1\}(z-x))\; |D\backslash chi\_E|(z)\; =\; \backslash int\_\{T\_x\}\; f(y)\; \backslash ,\; d\backslash mathcal\{H\}^\{n-1\}(y)$

for every continuous, compactly supported $f\; :\; \backslash R\; ^n\; \backslash to\; \backslash R$. In fact the subspace $T\_x$ is the orthogonal complement of the unit vector

:$\backslash nu\_E(x)\; =\; \backslash lim\_\{\backslash rho\; \backslash downarrow\; 0\}\; \backslash frac\{D\backslash chi\_E(B\_\backslash rho(x))\}\{|D\backslash chi\_E|(B\_\backslash rho(x))\}\; \backslash in\; \backslash R\; ^n$

defined previously. This unit vector also satisfies

:$\backslash lim\_\{\backslash lambda\; \backslash downarrow\; 0\}\; \backslash left\; \backslash \{\; \backslash lambda^\{-1\}(z\; -\; x)\; :\; z\; \backslash in\; E\; \backslash right\; \backslash \}\; \backslash to\; \backslash left\; \backslash \{\; y\; \backslash in\; \backslash R^n\; :\; y\; \backslash cdot\; \backslash nu\_E(x)\; >\; 0\; \backslash right\; \backslash \}$

locally in $L^1$, so it is interpreted as an approximate inward pointing unit normal vector to the reduced boundary $\backslash partial^*\; E$. Finally, $\backslash partial^*\; E$ is (n-1)-rectifiable and the restriction of (n-1)-dimensional Hausdorff measure $\backslash mathcal\{H\}^\{n-1\}$ to $\backslash partial^*\; E$ is $|D\backslash chi\_E|$, i.e.

:$|D\backslash chi\_E|(A)\; =\; \backslash mathcal\{H\}^\{n-1\}(A\; \backslash cap\; \backslash partial^*\; E)$ for all Borel sets $A\; \backslash subset\; \backslash R^n$.

In other words, up to $\backslash mathcal\{H\}^\{n-1\}$-measure zero the reduced boundary $\backslash partial^*\; E$ is the smallest set on which $D\backslash chi\_E$ is supported.

From the definition of the vector Radon measure $D\backslash chi\_E$ and from the properties of the perimeter, the following formula holds true:

:$\backslash int\_E\backslash operatorname\{div\}\backslash boldsymbol\{\backslash phi\}(x)\; \backslash ,\; \backslash mathrm\{d\}x\; =\; -\backslash int\_\{\backslash partial\; E\}\; \backslash langle\backslash boldsymbol\{\backslash phi\},\; D\backslash chi\_E(x)\backslash rangle$

\qquad \boldsymbol{\phi}\in C_c^1(\Omega, \R^n)

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 $\backslash partial^*\; E$ and the approximate inward pointing unit normal vector $\backslash nu\_E$. Precisely, the following equality holds

:$\backslash int\_E\; \backslash operatorname\{div\}\; \backslash boldsymbol\{\backslash phi\}(x)\; \backslash ,\; \backslash mathrm\{d\}x\; =\; -\; \backslash int\_\{\backslash partial^*\; E\}\; \backslash boldsymbol\{\backslash phi\}(x)\; \backslash cdot\; \backslash nu\_E(x)\; \backslash ,\; \backslash mathrm\{d\}\backslash mathcal\{H\}^\{n-1\}(x)\; \backslash qquad\; \backslash boldsymbol\{\backslash phi\}\; \backslash in\; C^1\_c(\backslash Omega,\; \backslash R^n)$

{{div col}}

• Geometric measure theory
• Divergence theorem
• Pfeffer integral

{{div col end}}

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|url = https://books.google.com/books?id=dNgsmArDoeQC&q=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

|last1 = Hudjaev

|first1 = 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&q=%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.

|author-link = 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 equations.

{{refend}}

• {{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