In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Caccioppoli set is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
whose
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
and has (at least
locally) a ''finite
measure''. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
is a
function of bounded variation.
History
The basic concept of a Caccioppoli set was first introduced by the Italian mathematician
Renato Caccioppoli
Renato Caccioppoli (; 20 January 1904 – 8 May 1959) was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.
Life a ...
in the paper : considering a plane set or a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
defined on an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in the
plane, he defined their
measure or
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
as the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
in the sense of
Tonelli of their defining
functions, i.e. of their
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s, ''provided this quantity was
bounded''. The ''measure of the
boundary of a set
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
was defined as a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
'', precisely a
set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R an ...
, for the first time: also, being defined on
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s, it can be defined on all
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s and its value can be approximated by the values it takes on an increasing
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
of
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s. Another clearly stated (and demonstrated) property of this functional was its ''
lower semi-continuity''.
In the paper , he precised by using a ''
triangular mesh'' as an increasing
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
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
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
, as expressed by the
Peano-Jordan Measure: ''to associate to every portion of a surface an
oriented
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
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
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
'' from a
subspace to the whole
ambient space
An ambient space or ambient configuration space is the space surrounding an object.
While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...
: the use of theorems generalizing the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
is frequently encountered in Caccioppoli research. However, the restricted meaning of
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
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
Lamberto Cesari (23 September 1910 – 12 March 1990) was an Italian mathematician naturalized in the United States, known for his work on the theory of surface area, the theory of functions of bounded variation, the theory of optimal control a ...
introduced the "right" generalization of
functions of bounded variation to the case of several variables only in 1936: 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 at the IV
UMI
Umi or UMI may refer to: Geography
* Umi, Iran, a village in Razavi Khorasan Province, Iran
* Umi, Fukuoka, a town in Japan People
* Umi-a-Liloa, king of the island of Hawaii
*Umi Dachlan, Indonesian female artist
* Umi Garrett, American female pi ...
Congress in October 1951, followed by five notes published in th
Rendicontiof the
Accademia Nazionale dei Lincei
The Accademia dei Lincei (; literally the "Academy of the Lynx-Eyed", but anglicised as the Lincean Academy) is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rom ...
. These notes were sharply criticized by
Laurence Chisholm Young
Laurence Chisholm Young (14 July 1905 – 24 December 2000) was a British mathematician known for his contributions to measure theory, the calculus of variations, optimal control theory, and potential theory. He was the son of William Henry ...
in the
Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.
The AMS also pu ...
.
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
Salzburg (, ; literally "Salt-Castle"; bar, Soizbuag, label=Austro-Bavarian) is the fourth-largest city in Austria. In 2020, it had a population of 156,872.
The town is on the site of the Roman settlement of ''Iuvavum''. Salzburg was founded ...
Congress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to a
mollifier
In mathematics, mollifiers (also known as ''approximations to the identity'') are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) f ...
, constructed from the
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
, 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. The same year he published his first paper on the topic i.e. : however, this paper and the closely following one did not attracted much interest from the mathematical community. It was only with the paper , reviewed again 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
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
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
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert F ...
and
Wendell Fleming
Wendell Helms Fleming (born March 7, 1928) is an American mathematician, specializing in geometrical analysis and stochastic differential equations.
Fleming received in 1951 his PhD under Laurence Chisholm Young at the University of Wisconsin� ...
published their paper , changing the approach to the theory. Basically they introduced two new kind of
currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
, respectively
normal currents and
integral currents: in a subsequent series of papers and in his famous treatise, Federer showed that Caccioppoli sets are normal
currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
of dimension
in
-dimensional
euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
s. However, even if the theory of Caccioppoli sets can be studied within the framework of theory of
currents
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
, 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
monograph
A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject.
In library cataloging, ''monogra ...
s in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
testify.
[See the "]References
Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name'' ...
" section.
Formal definition
In what follows, the definition and properties of
functions of bounded variation in the
-dimensional setting will be used.
Caccioppoli definition
Definition 1. Let ''
'' be an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
and let
be a
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
. The ''
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
of
in
'' is defined as follows
:
where
is the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
of
. That is, the perimeter of
in an open set
is defined to be the
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
of its
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
on that open set. If
, then we write
for the (global) perimeter.
Definition 2. The
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
is a Caccioppoli set if and only if it has finite perimeter in every
bounded open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
, i.e.
:
whenever
is open and bounded.
Therefore, a Caccioppoli set has a
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
whose
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
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
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
such that
:
As noted for the case of general
functions of bounded variation, this vector
measure is the
distributional or
weak
Weak may refer to:
Songs
* "Weak" (AJR song), 2016
* "Weak" (Melanie C song), 2011
* "Weak" (SWV song), 1993
* "Weak" (Skunk Anansie song), 1995
* "Weak", a song by Seether from '' Seether: 2002-2013''
Television episodes
* "Weak" (''Fear t ...
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of
. The total variation measure associated with
is denoted by
, i.e. for every open set
we write
for
.
De Giorgi definition
In his papers and ,
Ennio de Giorgi introduces the following
smoothing operator, analogous to the
Weierstrass transform
In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''.
Specifically, it is the function defin ...
in the one-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
case
:
As one can easily prove,
is a
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
for all
, such that
:
also, its
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is everywhere well defined, and so is its
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
:
Having defined this function, De Giorgi gives the following definition of
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
:
Definition 3. Let
be an
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of
and let
be a
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
. The ''
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
of
in
'' is the value
:
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 . 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
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
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
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
in
euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.
* If the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
of
is
, then
: this implies that if the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
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
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
and its
total variation measure
In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real ...
. 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
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
. There is an elementary lemma that guarantees that the
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
(in the sense of
distributions) of
, and therefore also
, is always contained in
:
Lemma. The support of the vector-valued Radon measure
is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the
topological boundary
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
of
.
Proof. To see this choose
: then
belongs to the
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ...
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
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ...
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 In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of the unit vector
:
defined previously. This unit vector also satisfies
:
locally in
, so it is interpreted as an approximate inward pointing
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (a ...
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
to the reduced boundary
. Finally,
is (n-1)-
rectifiable and the restriction of (n-1)-dimensional
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that ...
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
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
and from the properties of the perimeter, the following formula holds true:
:
This is one version of the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
for
domains with non smooth
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
. 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
:
See also
*
Geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
*
Divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
*
Pfeffer integral
Notes
References
Historical references
*. A paper surveying the history of the theory of sets of finite perimeter, from the seminal paper of
Renato Caccioppoli
Renato Caccioppoli (; 20 January 1904 – 8 May 1959) was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.
Life a ...
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.
*. The first paper containing the seminal concept of what a Caccioppoli set is.
*. The work where Caccioppoli made rigorous and developed the concepts introduced in the preceding paper .
*.The first paper detailing the theory of finite perimeter set in a fairly complete setting.
*. A selection from Caccioppoli's scientific works with a biography and a commentary of
Mauro Picone.
*. Available a
Numdam Cesari's watershed paper, where he extends the now called ''
Tonelli plane variation'' concept to include in the definition a subclass of the class of integrable functions.
*. The first note published by De Giorgi describing his approach to Caccioppoli sets.
*. The first complete exposition by De Giorgi of the theory of Caccioppoli sets.
*. The first paper of Herbert Federer illustrating his approach to the theory of perimeters based on the theory of currents.
*. A paper sketching the history of the theory of sets of finite perimeter, from the seminal paper of
Renato Caccioppoli
Renato Caccioppoli (; 20 January 1904 – 8 May 1959) was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.
Life a ...
to main discoveries.
Scientific references
*. An advanced text, oriented towards the theory of
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s in the multi-dimensional setting, written by one of the leading contributors.
*, 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
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
.
*, particularly Chapter 3, Section 14 "''Sets of Locally Finite Perimeter''".
*, 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
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
problem.
*, particularly part II, chapter 4 paragraph 2 "''Sets with finite perimeter''". One of the best books about –functions and their application to problems of
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, particularly
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
.
*; particularly chapter 6, "On functions in the space ". One of the best monographs on the theory of
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
.
*. A seminal paper where Caccioppoli sets and –functions are deeply studied and the concept of
functional superposition is introduced and applied to the theory of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s.
External links
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Function of bounded variationa
Encyclopedia of Mathematics
{{DEFAULTSORT:Caccioppoli Set
Mathematical analysis
Calculus of variations
Measure theory