Namespaces
Variants
Actions

Difference between revisions of "User:Camillo.delellis/sandbox"

From Encyclopedia of Mathematics
Jump to: navigation, search
 
(86 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
{{MSC|26A45}} (Functions of one variable)
+
{{MSC|49Q15|49Q20,49Q05,28A75,32C30,58A25,58C35}}
 +
[[Category:Classical measure theory]]
 +
{{TEX|done}}
 +
 
 +
An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century: Given a boundary wire, how does one find the (minimal) soap film which spans it? In mathematical terms: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also [[Plateau problem|Plateau problem]].) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, geometric analysis, harmonic analysis, and fractals.
  
−
{{MSC|26B30|28A15,26B15,49Q15}} (Functions of severable variables)
+
Many variational problems (cf. also [[Variational calculus|Variational calculus]]) are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged class. Much of the work in geometric measure theory has been directed towards placing this informal description on a formal footing appropriate for the study of surfaces.
  
−
[[Category:Analysis]]
+
=History=
  
−
{{TEX|done}}
+
=Measure theoretic concepts=
 +
One of the central issues of geometric measure theory is to define the concepts of volume, area
 +
and length in the uttermost generality. The roots of them are obviously in measure theory. In particular, the concept of [[Outer measure|outer measure]] in a [[Metric space|metric
 +
space]] plays a fundamental role.
  
−
==Functions of one variable==
+
==Caratheodory construction==
−
===Classical definition===
+
The following is a common construction of [[Metric outer measure|metric outer measures]] in metric spaces $(X, d)$.
−
Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if
 
−
its [[Variation of a function|total variation]] is bounded. The total variation is defined in the following way.
 
  
 
'''Definition 1'''
 
'''Definition 1'''
−
Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$,
+
If  $\mathcal{C}$ is a class of subsets of $X$ containing the empty set,  $\nu  : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset)  =0$ and $\delta\in ]0, \infty]$, then for every $A\subset X$ we define
−
where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by
+
\[
−
\begin{equation}\label{e:TV}
+
\mu^\delta  (A) := \inf \left\{ \sum_i \nu (E_i) : \{E_i\}_{i\in \mathbb N}  \subset  \mathcal{C} \mbox{ covers } A \mbox{ and } {\rm diam}\, (E_i) \leq  \delta\,\right\}\,  
−
TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\,
+
\]
−
\end{equation}
+
and  
−
(cp. with Section 4.4 of {{cite|Co}} or Section 10.2 of {{Cite|Ro}}).
 
−
 
 
−
====Generalizations====
 
−
The definition of total variation of a function of one real variable can be easily generalized when the target is a [[Metric space|metric space]] $(X,d)$: it suffices to substitute $|f(b_i)-f(a_i)|$ with $d (f(a_i), f(b_i))$ in \ref{e:TV}. Consequently, one defines functions of bounded variation taking values in an arbitrary metric space. Observe that, if $f:I\to X$ is a function of bounded variation and $\varphi:X\to Y$ a [[Lipschitz condition|Lipschitz map]], then $\varphi\circ f$ is also a function of bounded variation and
 
 
\[
 
\[
−
TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, ,
+
\mu (A) := \lim_{\delta\downarrow 0}\; \mu^\delta (A)\, .
 
\]
 
\]
−
where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.
 
  
−
As a corollary we derive
+
Observe that the latter limit exists because $\mu^\delta (A)$ is a nonincreasing function of $\delta$.
 +
This construction is often called ''Caratheodory construction''. See Section 2.1 of {{Cite|KP}} (cp. also with {{Cite|Fe}}).
  
−
'''Proposition 2'''
+
'''Theorem 2'''
−
A function $(f^1, \ldots, f^k) = f: I\to \mathbb R^k$ is of bounded variation if and only if each coordinate function $f^j$ is of bounded variation.
+
Let $\nu$ and $\mu$ be as in Definition 7. Then $\mu$ is a metric outer measure. Thus there is a [[Algebra of sets|$\sigma$-algebra]] $\mathcal{A}$ which
 +
contains the [[Borel set|Borel sets]] and such that the restriction of $\mu$ to $\mathcal{A}$ is $\sigma$-additive.
  
−
===General properties===
+
(Cp.  with Claims 1,2 and 3 in the proof of Theorem 1 in Section 2.1 of {{Cite|EG}}: although the reference handles the cases of Hausdorff outer  measures, the proof extends verbatim to the setting above).
−
====Jordan decomposition====
 
−
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
 
  
−
'''Theorem 3'''
+
===Hausdorff measures===
−
Let $I\subset \mathbb R$ be an interval. A function $f: I\to\mathbb R$ has bounded variation if and only if it can be written as the difference of two bounded nondecreasing functions.
+
The Caratheodory construction gives several generalizations of the concept of dimension and
 +
volume. The most common is due to Hausdorff (cp. with {{Cite|Ha}}).
  
−
(Cp. with Theorem 4 of Section 5.2 in {{Cite|Ro}}). Indeed it is possible to find a canonical representation of any function of bounded variation as difference of nondecreasing functions.
+
'''Definition 3'''
−
 
+
The  [[Hausdorff measure|Hausdorff outer measure]] $\mathcal{H}^\alpha$ is given by such $\mu$ as in  Definition 1 when we choose $\mathcal{C} = \mathcal{P} (X)$ and $\nu  (A) = \omega_\alpha ({\rm diam}\, (A))^\alpha$, where
−
'''Theorem 4'''
 
−
If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$. The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.
 
−
 
 
−
(Cp. with Theorem 3 of Section 5.2 in {{Cite|Ro}}). The latter representation of a function of bounded variation is also called [[Jordan decomposition]].
 
−
 
 
−
====Continuity====
 
−
It follows immediately from Theorem 3 that
 
−
 
 
−
'''Proposition 5'''
 
−
If $f:I\to [a,b]$ is a function of bounded variation, then
 
−
* The right and left limits
 
 
\[
 
\[
−
f (x^+) :=\lim_{y\downarrow x} f (y) \qquad f (x^-):= \lim_{y\uparrow x} f(y)
+
\omega_\alpha = \frac{\pi^{\alpha/2}}{\Gamma \left(\frac{\alpha}{2}+1\right)}\,
 
\]
 
\]
−
exist at every point $x\in I$;
+
(cp. with Section 2.1 of {{Cite|EG}}).
−
* The set of points of discontinuity of $f$ is at most countable.
 
  
−
'''Warning 6''' However, according to the definitions given above, it may happen that at a goven point right and left limits coincide, but nonetheless the function $f$ is discontinuous. For instance the function $f:\mathbb R\to\mathbb R$ given by
+
When $\alpha$ is a (positive) integer $n$, $\omega_n$ equals the Lebesgue measure of the unit ball in $\mathbb R^n$. With
−
\[
+
this choice the $n$-dimensional Hausdorff outer measure on the euclidean space $\mathbb R^n$ coincides
−
f (x) =\left\{\begin{array}{ll}
+
with the Lebesgue measure. It must be noted, however, that some authors prefer to set the constant $\omega_\alpha$ equal to $1$ (see for instance {{Cite|Ma}}).
−
1 \qquad &\mbox{if $x=0$}\\
 
−
0 \qquad &\mbox{otherwise}
 
−
\end{array}\right.
 
−
\]
 
−
is a function of bounded variation
 
  
−
====Precise representative====
+
====Hausdorff dimension====
−
In order to avoid patologies as in '''Warning 6''' it is customary to postulate some additional assumptions for functions of bounded variations. Two popular choices are
+
The following is a simple consequence of the definition (cp. with Theorem 4.7 of {{Cite|Ma}}).
−
* the imposition of right (resp. left) continuity, i.e. at any point $x$ we impose $f(x)=f (x^+)$ (resp. $f(x)=f(x^-$), cp. with Section 4.4 of {{Cite|Co}};
 
−
* at any point $x$ we impose $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.
 
−
The latter is perhaps more popular because of the [[Jordan criterion]] (see '''Theorem 11''' below) and it is often called ''precise representative''.  
 
  
−
====Differentiability====
+
'''Theorem 4'''
−
Functions of bounded variation of one variable are classically differentiable at a.e. point of their domain of definition, cp. with Corollary 5 of Section 5.2 in {{Cite|Ro}}. It turns out that such derivative is always a summable function (see below in the section '''Structure theorem'''). However, the fundamental theorem of calculus does not apply in this case, i.e. there are continuous functions $f:[a,b]\to\mathbb R$ of bounded variation such that the identity
+
For $0\leq s<t<\infty$ and $A\subset X$ we have
−
\[
+
* $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
−
f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt
+
* $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.
−
\]
 
−
fails for a set of pairs $(b', a')\in I\times I$ of positive measure (see below in the section '''Examples''').
 
−
 
 
−
===Measure theoretic characterization===
 
−
Classically right-continuous functions of bounded variations can be mapped one-to-one to [[Signed measure|signed measures]]. More precisely, consider a signed measure $\mu$ on (the [[Borel set|Borel subsets ]] of) $\mathbb R$ with finite total variation (see [[Signed measure]] for the definition). We then define the function
 
−
\begin{equation}\label{e:F_mu}
 
−
F_\mu (x) := \mu (]-\infty, x])\, .
 
−
\end{equation}
 
−
 
 
−
'''Theorem 7'''
 
−
* For every signed measure $\mu$ with finite total variation, $F_\mu$ is a right-continuous function of bounded variation such that $\lim_{x\to -\infty} F_mu (x) = 0$ and $TV (f)$ equals the total variation of $|\mu|$.
 
−
* For every right-continuous function $f:\mathbb R\to \mathbb R$ of bounded variations such that $\lim_{x\to-\infty} f (x) = 0$ there is a unique signed measure $\mu$ such that $f=F_\mu$
 
−
 
 
−
For a proof see Section 4 of Chapter 4 in {{Cite|Co}}. Obvious generalizations hold in the case of different domains of definition.
 
−
 
 
−
====Distributional derivatives: modern definition====
 
−
The measure $\mu$ is indeed the [[Generalized derivative|generalized derivatie]] of the function $f=F_\mu$ in the sense of distributions. More precisely
 
−
\begin{equation}\label{e:distrib}
 
−
\int f(t)\varphi' (t)\, dt = -\int \varphi (t)\, d\mu (t) \qquad \forall \varphi\in C^\infty_c (\mathbb R)\, .
 
−
\end{equation}
 
−
This identty is the starting point for the modern definition of functions of bounded variation, cp. with {{Cite|AFP} or Chapter 5 of {{Cite|EG}}.
 
  
−
'''Definition 8'''
+
The [[Hausdorff dimension]] ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as
−
Let $I\subset\mathbb R$ be a bounded open interval. A function $f\in L^1 (E)$ is said to be of bounded variation if
 
−
\begin{equation}\label{e:variation_modern}
 
−
\sup \left\{ \int \varphi' (t) f(t)\, dt \;:\; \varphi\in C^\infty_c (I), \|\varphi\|_{C^0} \leq 1\right\} <\infty\, .
 
−
\end{equation}
 
  
−
The following theorem links the classical and the modern definitions. See section 3.2 of {{Cite|AFP}} for a proof.
+
'''Definition 5'''
 +
\begin{align*}
 +
{\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\
 +
&=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, .
 +
\end{align*}
  
−
'''Theorem 9'''
+
===Other measures related to the volume===
−
Let $f$ and $I$ be as in Definition 8. Then there is a function $\tilde{f}:I\to\mathbb R$ and a signed measure $\mu$ on $I$ such that
+
The Caratheodory construction can be used to build other generalizations of the concept of volume, such as
−
* $\mu$ is the derivative, in the sense of distributions, of $f$, i.e. \eqref{e:distrib} holds
+
* The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of {{Cite|KP}}.
−
* $F_\mu = \tilde{f} = f$ almost everywhere
+
* The Gross outer measures, the Caratheodory outer measures, the  integral-geometric outer measures (see also [[Favard measure]]) and the Gillespie outer measures in $\mathbb R^n$, see Sections  2.1.3-2.1.4-2.1.5 of {{Cite|KP}} (cp. also  with 2.10.2-2.10.3-2.10.4 of {{Cite|Fe}}).
−
* $\tilde{f}$ is a function of bounded variation in the sense of '''Definition 1'''
+
All these measures coincide on sufficiently regular set (in particular on [[Rectifiable set|rectifiable sets]], see below), but they are, in general, different.
−
* $TV (\tilde{f})$ equals the total variation of the measure $\mu$ which in turn is equal to the supremum in \eqref{e:variation_modern}.
+
===Fractals===
 +
Fractals have been originally defined by B.B. Mandelbrot as point sets with non-integer Hausdorff dimension, although this name is nowadays also applied to sets of integer dimensions which are [[Purely unrectifiable set|purely unrectifiable]]. Classical examples of fractals are the [[Cantor set]], the [[Koch curve]] and the [[Julia set]] of an holomorphic function $f$. Fractals are typically self-similar.  
  
−
Similar definitions and properties can be given for more general domains. However some caution is needed for unbounded domains since then functions of bounded variation are, in general, only '''locally''' summable.
+
Sets with non-integer Hausdorff dimension have been objects of study of geometric measure theory since the pioneering works of Besicovitch and Marstrand. For an account of the modern mathematical theory of fractals we refer the reader to {{Cite|Fa2}}.
  
−
===Structure theorem===
+
==Differentiation theorem==
−
It is possible to relate the pointwise properties of a function $f: I\to \mathbb R$ of bounded variation with the properties of its generalized derivative $\mu$. More pecisely, using the [[Radon-Nikodym decomposition]] we write $\mu = g \lambda + \mu_s$, where $\mu_s$ is a singular measure with respect to the Lebesgue measure $\mu$. We further follow the discussion of Section 3.2 of {{Cite|AFP}} and decompose $\mu_s = \mu_c +\mu_j$, where $\mu_c$ is the ''non-atomic'' part of the measure $\mu_s$, i.e.
+
It is common in geometric measure theory to construct measures as solutions to geometric problems or to partial differential equations. In these cases measures are often considered as a suitable relaxation of $k$-dimensional surfaces. For instance, given a $k$-dimensional surface $\Gamma\subset \mathbb R^n$, one can natural
−
\[
+
associate to this surface the measure $\mu (A) := \mathcal{H}^k (A\cap \Gamma)$. It is therefore of interest to understand which assumptions guarantee that a measure has a structure as in the latter example.
−
\mu_c (\{x\}) = 0\qquad \mbox{for every $x\in I$}\,
+
A fundamental tool in this direction, which is used ubiquitously in geometric measure theory, is the following theorem (cp. with [[Differentiation of measures]]), usually credited to Besicovitch (see {{Cite|Be4}} and {{Cite|Be5}}) and which  
−
\]
+
gives an explicit characterization of the [[Absolutely continuous measures|Radon-Nykodim decomposition]] for locally finite [[Radon  measure|Radon measures]] on the euclidean space.
−
and $\mu_j$ is the purely atomic part of $\mu_s$, that is, there is a set $J$ at most countable and weights $c_x\in \mathbb R, x\in J$ such that
 
−
\[
 
−
\mu_j (E) = \sum_{x\in J\cap E} c_x\, .
 
−
\]
 
−
If we denote by $\delta_x$ the [[Delta-function|Dirac mass]] at the point $x$, then $\mu_j = \sum_{x\in J} c_x \delta_x$. We then have the following theorem (cp. with Section 3.2 of {{Cite|AFP}}), which is often referred to as ''BV structure theorem'' fur functions of one variable.
 
  
−
'''Theorem 10'''
+
'''Theorem 6''' (cp. with Theorem 2.12 of {{Cite|Ma}} and Theorem 2 in Section 1.6 of {{Cite|EG}})
−
Let $I = ]a,b[$, $f:I\to \mathbb R$ a right-continuous function of bounded variation and $\mu = g\lambda + \mu_c + \mu_j$ its generalized derivative.  
+
Let $\mu$ and $\nu$ be two locally finite Radon measures on $\mathbb R^n$. Then,
−
* If $J$ denotes the set of points of discontinuity of $f$, then
+
* the limit
 
\[
 
\[
−
\mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, .
+
f(x) := \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))}
 
\]
 
\]
−
* At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f(x) = g(x)$.
+
exists at $\mu$-a.e. $x$ and defines a $\mu$-measurable map;
−
 
+
* the set
−
====Lebesgue decomposition====
+
\begin{equation}\label{e:singular}
−
Observe also that, if we define the functions
+
S:= \left\{ x: \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} = \infty\right\}
−
* $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
 
−
* $f_j (x) := \mu_j (]a, x])$,
 
−
* $f_c (x) := \mu_c (]a, x])$,
 
−
then
 
−
* $f_a$ is an [[Absolute continuity|absolutely continuous function]]
 
−
* $f_c$ is a [[Singular function|singular function]]
 
−
* $f_j$ is a [[Jump function|jump function]].
 
−
Then $f=f_a+f_c+f_j$ is called the [[Lebesgue decomposition]] of the function $f$ and it is unique up to constants. For such funct
 
−
 
 
−
===Examples===
 
−
====Smooth functions====
 
−
If $f: I\to\mathbb R$ is smooth, then we have the identity
 
−
\begin{equation}\label{e:smooth_var}
 
−
TV (f) = \int_I |f'(t)|\, dt\, .
 
 
\end{equation}
 
\end{equation}
−
 
+
is $\nu$-measurable and a $\mu$-null set;
−
====Absolutely continuous functions====
+
* $\nu$ can be decomposed as $\nu_a + \nu_s$, where
−
[[Absolute continuity|Absolutely continuous]] functions are functions of bounded variation and indeed they are the largest class of functions of bounded variation for which \eqref{e:smooth_var} hold. Indeed absolutely continuous functions can be characterized as those functions of bounded variation such that their generalized derivative is an [[Absolute continuity|absolutely continuous measure]].
 
−
 
 
−
====Jump functions====
 
−
The indicator function of the half line, also called [[Heaviside function]]
 
 
\[
 
\[
−
{\bf 1}_{[a, \infty[} (x) :=  
+
\nu_a (E) = \int_E f\, d\mu
−
\left\{\begin{array}{ll}
 
−
0 \qquad &\mbox{if $x<a$}\\
 
−
1 \qquad &\mbox{if $x\geq a$}
 
−
\end{array}\right.
 
 
\]
 
\]
−
is a function of bounded variation (on $\mathbb R$) with total variation equal to $1$. Its generalized derivative is the [[Delta-function|Dirac mass] $\delta_a$. Obviously the Heaviside function is differentiable a.e. with derivative $0$ but its total variationis $1$, thereby showing that \eqref{e:smooth_var} fails for general functions of bounded variation.
+
and
−
 
 
−
The Heaviside function is a prototype of [[Jump function|jump function]] in the sense of the [[Lebesgue decomposition]]. If $f$ is a jump function on $\mathbb R$ with $\lim_{x\to\infty} f(x) = 0$, then there are two (at most) countable collections $\{c_i\}, \{a_i\}\subset\mathbb R$ such that
 
 
\[
 
\[
−
f = \sum_i c_i {\bf 1}_{[a_i, \infty[}\, .
+
\nu_s (E) = \nu (S\cap E)\, .
 
\]
 
\]
−
 
+
Moreover, for $\mu$-a.e. $x$ we have:
−
====Cantor ternary function====
+
\begin{equation}\label{e:Lebesgue}
−
The [[Cantor ternary function]], also called Devil's staircase (and Cantor-Vitali functions, by some Italian authors) is the most famous example of a continuous function of bounded variation for which \eqref{e:smooth_var} fails (which was first pointed out by Vitali in {{Cite|Vi}}). In fact it is a nondecreasing function such that its derivative vanishes almost everywhere. Its generalized derivative $\mu$ vanishes on the complement of the [[Cantor set]] and the function is the prototype of [[Singular function|singular function]] in the [[Lebesgue decomposition]].
+
\lim_{r\downarrow  0} \frac{1}{\mu (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, d\mu (y) = 0\qquad \mbox{and}\qquad \lim_{r\downarrow 0} \frac{\nu_s (B_r (x))}{\mu  (B_r (x))}= 0\, .
−
 
 
−
===Historical remark===
 
−
Functions of bounded variation were introduced for the first time by C. Jordan in {{Cite|Jo}} to study the  pointwise convergence of Fourier series. In particular Jordan proved the following generalization of [[Dirichlet theorem]] on the convergence of Fourier series, called [[Jordan criterion]]
 
−
 
 
−
'''Theorem 11'''
 
−
Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic summable function.
 
−
* If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
 
−
* If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$.
 
−
 
 
−
For a proof see Section 10.1 and Exercises 10.13 and 10.14 of {{Cite|Ed}}. The criterion is also called Jordan-Dirichlet test, see {{Cite|Zy}}
 
−
 
 
−
==Functions of several variables==
 
−
===Historical remarks===
 
−
After the introduction by Jordan of functions of bounded variations of one real variable, several authors attempted to generalize the concept to functions of more than one variable. The first attempt was made by Arzelà and Hardy in 1905, see [[Arzelà variation]] and [[Hardy variation]], followed by Vitali, Fréchet, Tonelli and Pierpont, cp. with [[Vitali variation]], [[Fréchet variation]], [[Tonelli plane variation]] and [[Pierpont variation]] (moreover, the definition of Vitali variation was also considered independently by Lebesgue and De la Vallée-Poussin). However, the point of view which became popular and it is nowadays accepted in the literature as most efficient generalization of the $1$-dimensional theory is due to De Giorgi and Fichera (see {{Cite|DG}} and {{Cite|Fi}}). Though with different definitions, the functions of bounded variation defined by De Giorgi and Fichera are equivalent (and very close in spirit) to the ''distributional theory'' described below. A promiment role in the further developing of the theory was also played by Fleming and Federer. Moreover, Krickeberg and Fleming showed, independently, that the current definition of functions of bounded variation is indeed equivalent to a slight modification of Tonelli's one {{Cite|To}}, proposed by Cesari {{Cite|Ce}}, cp. with the section '''Tonelli-Cesari variation''' below. We refer to Section 3.12 of {{Cite|AFP}} for a thorough discussion of the topic.
 
−
====Link to the theory of currents====
 
−
Functions of bouned variation in $\mathbb R^n$ can be identified with $n$-dimensional [[Current|currents]] in $\mathbb R^n$. This is the point of view of Federer, {{Cite|Fe}}, which thus derives most of the conclusions of the theory of $BV$ functions as special cases of more general theorems for normal currents,
 
−
===Definition===
 
−
Following Section 3.1 of {{Cite|AFP}},
 
−
 
 
−
'''Definition 12'''
 
−
Let $\Omega\subset \mathbb R^n$ be open. $u\in L^1 (\Omega)$ is a function of ''bounded variation'' if the [[Generalized derivative|generalized partial derivatives]] of $u$ in the sense of distributions are [[Signed measure|signed measures]], i.e. if for every $i\in \{1, \ldots, n\}$ there is a signed measure $\mu_i$ (with finite total variation) on the [[Algebra of sets|$\sigma$-algebra]] of [[Borel set|Borel sets]] of $\Omega$ such that
 
−
\begin{equation}\label{e:distrib2}
 
−
\int_\Omega u \frac{\partial \varphi}{\partial x_i}\, d\lambda = - \int_\Omega \varphi\, d\mu_i \qquad \forall \varphi\in C^\infty_c (\Omega)\, .
 
−
\end{equation}
 
−
The vector measure $\mu := (\mu_1, \ldots, \mu_n)$ will be denoted by $Du$ and its variation measure (see [[Signed measure]] for the definition) will be denoted by $|Du|$.
 
−
The vector space of all functions of bounded variations on $\Omega$ is denoted by $BV (\Omega)$.
 
−
 
 
−
We assume $u\in L^1 (\Omega)$ to keep the technicalities at a minimum. However, it is possible to relax this assumption, as it is possible to define the spacel $BV_{loc} (\Omega)$ of functons of bounded local variation, i.e. such that $u\in BV (\Gamma)$ for every open $\Gamma\subset\subset\Omega$ (see {{Cite|AFP}}).
 
−
 
 
−
====Total variation====
 
−
Some authors use instead the following alternative road (cp. with Section 5.1 of {{Cite|EG}}).
 
−
 
 
−
'''Definition 13'''
 
−
Let $\Omega\subset \mathbb R^n$ be open. The total variation of $u\in L^1 (\Omega)$ is given by
 
−
\begin{equation}\label{e:diverg}
 
−
V (u, \Omega) := \sup \left\{ \int_\Omega u\, {\rm div}\, \psi : \psi\in C^\infty_c (\Omega, \mathbb R^n), \, \|\psi\|_{C^0}\leq 1\right\}\, .
 
 
\end{equation}
 
\end{equation}
  
−
As a consequence of the [[Radon-Nikodym theorem]] we then have
+
===Covering arguments===
 +
Theorem 6 does not hold in general metric spaces. It holds provided the metric  space satisfies some properties about covering of sets with balls (cp. with  [[Covering theorems (measure theory)]]). In fact, aside from their links to the differentiation of measures, both the Vitali and Besicovitch covering Lemmas (see again [[Covering theorems (measure theory)]]) and similar arguments play a pivotal role in several fundamental results of geometric measure theory.
  
−
'''Prposition 14'''
+
=Rectifiable sets=
−
A function $u\in L^1 (\Omega)$ is a function of bounded variation if and only if $V(u, \Omega)<\infty$ and moreover $V (u,\Omega) = |Du| (\Omega)$.
+
Also called ''countably rectifiable sets'', rectifiable sets are central objects of study in  [[Geometric measure theory]], cp. with [[Rectifiable set]]. Rectifiable sets of the  euclidean space are fairly close to $C^1$ submanifolds, a sort of measure-theoretic counterpart of the latter. As such they  have an integer Hausdorff dimension. In the special case of $1$-dimensional sets of the euclidean space, they were first introduced by Besicovitch, as a suitable generalization of [[Rectifiable curve|rectifiable curves]].
 +
In what follows we  will use the terminology ''$m$-dimensional rectifiable set''. Some  authors prefer the terminology ''countably $m$-rectifiable set'' or, briefly, ''$m$-rectifiable''.
  
−
====Consistency with the one variable theory====
+
Rectifiable subsets of the Euclidean space $\mathbb R^n$ can be defined in several ways, see [[Rectifiable set]]. We adopt here the following one
−
By '''Theorem 9''', '''Definition 13''' is consistent, in the case $n=1$, with '''Definition 1'''. More precisely, if $I\subset \mathbb R$ is a bounded open interval and $f:I\to \mathbb R$ a right-continuous $L^1$ function, then $V(f, I) = TV (f)$ (in particular, if $TV (f)<\infty$, then necessarily $f\in L^1 (I)$ and $V (f, I)<\infty$). Viceversa, if $f\in L^1 (I)$ and $V(f, I)$, then there is a right-continuous function $\tilde{f}$ which coincides $\lambda$-a.e. with $f$ and such that $TV (\tilde{f}) = V (f, I)$. Similar assertions can be proved for more general intervals. However some technical adjustments are needed if the domain is unbounded because a function of bounded variation in the sense of '''Definition 1''' is not necessarily summable.
 
  
−
====Generalizations====
+
'''Definition 7''' (cp. with Lemma 11.1 of {{Cite|Si}})
−
Let $\Omega\subset \mathbb R^n$ be an open set. $f:\Omega\to\R^m$ belongs to the space $BV (\Omega, \mathbb R^m)$ if each component function is an element in $BV (\Omega)$. A far-reaching generalization for general metric targets has been introduced
+
A Borel  set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if  it  has Hausdorff dimension $k$ and there is a countable family of  $C^1$ $k$-dimensional submanifolds of $\mathbb R^n$ which cover  $\mathcal{H}^k$-almost all $E$.  
−
by Ambrosio in {{Cite|Am}}:
 
  
−
'''Definition 14'''
+
The assumption that $E$ is a Borel set might be dropped, but in this case the set might not be $\mathcal{H}^k$-measurable (see [[Rectifiable set]]). In what follows we will assume that rectifiable sets $E$ are $\mathcal{H}^k$ measurable: $\sigma$-finite $\mathcal{H}^k$-measurable sets can be decomposed into the union of a Borel set and an $\mathcal{H}^k$-null set.
−
Let $\Omega\subset \mathbb R^n$ be a bounded set and $(X,d)$ a metric space. A Lebesgue measurable map $f:\Omega \to X$ is a generalized function of bounded variation if
 
−
* $\varphi\circ f\in BV (\Omega)$ for every Lipschitz function $\varphi:X\to\mathbb R$. 
 
−
* There is a measure $\mu$ such that $|D (\varphi\circ f)|\leq {\rm Lip}\, (\varphi) \mu$ for every Lipschitz function $\varphi:X\to\mathbb R$.
 
  
−
This definition, which found recently quite important applications, is consistent with the one-dimensional theory and with the case $X=\mathbb R^m$ given above (for the latter see the section '''Volpert chain rule''').
+
==Besicovitch's works==
 +
As already mentioned, the theory of rectifiable $1$-dimensional sets was first laid down by Besicovitch in a series of works (see {{Cite|Be1}}, {{Cite|Be2}}, {{Cite|Be3}}). In these seminal papers Besicovitch considered and solved several questions which became later pivotal for rectifiable sets of higher dimension. He also introduced many useful tools (such as the Besicovitch covering theorem, see [[Covering theorems (measure theory)]]) which have been widely used in different contexts.
 +
===One-dimensional sets===
 +
The theory of one-dimensional rectifiable sets is somewhat special  since much stronger theorems can be proved which fail for higher  dimensions. Perhaps the most useful one is the following (cp. with  Theorem 3.14 of {{Cite|Fa}}):
  
−
===Functional properties===
+
'''Theorem 8'''
−
The space $BV (\Omega)$ enjoys several properties that are typical of the [[Sobolev space|Sobolev spaces]] $W^{1,p} (\Omega)$.  
+
A  [[Continuum|continuum]], i.e. a compact connected set, $E\subset\mathbb  R^n$ of finite $\mathcal{H}^1$ measure is always rectifiable and arcwise connected. Indeed it is always the image of a [[Rectifiable curve|rectifiable curve]].
−
====Banach space structure====
 
−
The norm $\|u\|_{BV} := \|u\|_{L^1} + V (u, \Omega)$ endows $BV (\Omega)$ with a [[Banach space]] structure. $BV (\Omega)$ is not reflexive but it is the dual of a separable space (see Remark 3.12 of Section 3.1 in {{Cite|AFP}}).
 
−
$BV (\Omega)$ contains $W^{1,1} (\Omega)$ and the norm $\|\cdot\|_{BV}$ restricted to $W^{1,1}$ coincides with the  $\|\cdot\|_{W^{1,1}}$ norm. In fact $W^{1,1} (\Omega)$ is a closed subspace of $BV (\Omega)$ (see Example 1 of Section 5.1 in {{Cite|EG}}).
 
  
−
====Semicontinuity of the variation====
+
We refer to {{Cite|Fa}} for a comprehensive account of the theory of rectifiable one-dimensional sets.
−
If a sequence of functions $\{u_n\}\in L^1 (\Omega)$ converges strongly to $L^1 (\Omega)$, then
 
−
\[
 
−
\liminf_{n\to\infty}\, V (u_n, \Gamma)\geq V (u, \Gamma)
 
−
\]
 
−
for every open set $\Gamma\subset\Omega$ (cp. with Remark 3.5 of {{Cite|AFP}}). In particular, if $\liminf\, V (u_n,\Omega)<\infty$, then $u\in BV (\Omega)$.
 
−
====Approximation with smooth functions====
 
−
'''Theorem 15'''
 
−
A function $u$ belongs to $BV (\Omega)$ if and only if there exists a sequence of smooth functions $\{u_n\}$ such that
 
−
* $\|u_n-u\|_{L^1 (\Omega)} \to 0$
 
−
* $\liminf_n V (u_n, \Omega) < \infty$.
 
−
Moreover, for every $u\in BV (\Omega)$ there is an approximating sequence $\{u_n\}\in C^\infty\cap BV (\Omega)$ converging strongly to $u$ and such that $V (u_n, \Omega)\to V (u, \Omega)$ (therefore $\|u_n\|_{BV}\to \|u\|_{BV}$.
 
  
−
Cp. with Theorem 3.9 of Section 5.1 in {{Cite|AFP}}.
+
==General dimension and codimension==
−
However, differently from the usual Sobolev spaces, the space $C^\infty (\Omega)$ is ''not dense'' in the strong topology: its closure is instead $W^{1,1} (\Omega)$.
+
The main importance of the class of rectifiable sets is that it possesses many of the nice properties of the smooth surfaces which one is seeking to generalize. For example, although, in general, classical tangents may not exist, an $m$-rectifiable set will possess a unique approximate tangent at $\mathcal{H}^m$-almost every point (see [[Rectifiable set]]).
−
====Weak$^\star$ convergence====
 
−
A sequence $\{u_n\}$ converges weakly$^\star$ in $BV (\Omega)$ to $u$ if $u_n\to u$ strongly in $L^1 (\Omega)$ and $Du_h$ converges weakly$^\star$ in the sense of measures to $Du$ (cp. with [[Convergence of measures]]). In fact a sequence converges weakly$^\star$ if and only if it converges in $L^1$ and it is bounded in the $BV$ norm (cp. with Proposition 3.13 of Section 3.1 in {{Cite|AFP}}
 
  
−
In fact, closed and bounded convex subsets of $BV (\Omega)$ are weakly$^\star$ compact (cp. with Theorem 3.23 in Section 3.1 of {{Cite|AFP}}).
+
===Besicovitch-Federer projection theorem===
−
====Extension theorems====
+
Often, one is faced with the task of showing that some set, which is a solution to the problem under investigation, is in fact rectifiable, and hence possesses some smoothness. A major concern in geometric measure theory is finding criteria which guarantee rectifiability, several of these criteria are listed in [[Rectifiable set]]. One of the most striking results in this direction is the Besicovitch–Federer projection theorem, which illustrates the stark difference between rectifiable and unrectifiable sets. This theorem characterize purely unrectifiable $m$-dimensional sets as those sets whose projections are $\mathcal{H}^m$-negligible on almost every $m$-dimensional plane (for the precise statement, we refer to [[Rectifiable set]]). This deep result was first proved for $1$-dimensional sets in the plane by A.S. Besicovitch, and later extended to higher dimensions by H. Federer. B. White in {{Cite|Wh}} has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version.
−
If $\Omega$ is an open set with compact Lipschitz boundary, then any function $u\in BV (\Omega)$ can be extended to a function $u\in BV (\mathbb R^n)$ (cp with Theorem 3.21 of Section 3.1 in {{Cite|AFP}}). Not all bounded open subsets possess this extension property: however the class of extension domains is larger than the class of open sets with compact Lipschitz boundary.
 
−
====Sobolev inequality====
 
−
The usual [[Sobolev inequality]] which holds for $W^{1,1}$ functions extends to $BV$ functions as well. Namely, there are constants $C(n)$ depending only on $n\in\mathbb N\setminus \{0\}$ such that:
 
−
*$\|f\|_{L^\infty}\leq C(1) TV (f)$ for any $f\in BV (\mathbb R)$;
 
−
*$\|f\|_{L^{n/(n-1)}}\leq V (u,\mathbb R^n)$ for any $f\in BV (\mathbb R^n)$ for any $n\geq 2$.
 
−
In the case $n=1$ the optimal constant is indeed $C(1)=1$ and the inequality follows easily from the considerations in the section '''Measure theoretic characterization'''. For the case $n\geq 2$ we refer to Theorem 1 of Section 5.6 in {{Cite|EG}} or Theorem 3.47 of Section 3.4 of {{Cite|AFP}}). The Sobolev inequality combined with the extension theorems give the embeddings $BV (\Omega)\subset L^p (\Omega)$ for any extension domain $\Omega$ and every $p\in [1, \frac{n}{n-1}]$. Such embedding is compact if $\Omega$ is bounded and $p<\frac{n}{n-1}$ (cp. with Corollary 3.49 of {{Cite.
 
−
====Poincaré inequality====
 
−
The usual [[Poincaré inequality]] for $W^{1,1}$ extends as well to $BV$ functions., Namely, there is a constant $C(n)$ such that, for $n\geq 2$,
 
−
\[
 
−
\left(\int_{B_r (x)} |u (y)-\bar{u}|^{\frac{n-1}{n}}\right)^{\frac{n-1}{n}}\, \;\leq\; C (n) \, V (u, B_r (x)) \qquad \mbox{for every $u\in BV (B_r (x))$}
 
−
\]
 
−
where $\bar{u}$ denotes the average of $u$ on $B_r (x)$ (and $B_r (x)\subset \mathbb R^n$ is the open ball
 
−
with radius $r$ and center $x$). See Theorem 1 of Section 5.6 in {{Cite|EG}} or Remark 3.50 of Section 3.4 on {{Cite|AFP}}. In fact such inequalities hold also on more general domains $\Omega$, with constants depending on the
 
−
specific geometry of $\Omega$.
 
−
====Trace operator====
 
−
For functions of bounded variations a suitable extension of the classical theory of traces of Sobolev spaces holds as well. In what follows we denote by $\mathcal{H}^{n-1}$ the [[Hausdorff measure|Hausdorff $n-1$-dimensional measure]].
 
  
−
'''Theorem 16'''
+
===Recitifiable measures===
−
Assume $\Omega$ is open and bounded, with $\partial \Omega$ of class $C^1$. Then there exists a bounded linear mapping
+
It is also possible (and useful) to define a notion of rectifiability  for (locally finite) Radon measures: A [[Radon measure|Radon measure]] $\mu$ is said to be $m$-rectifiable if it  is absolutely continuous (cf. also [[Absolute continuity|Absolute  continuity]]) with respect to the $m$-dimensional  Hausdorff measure and there is an $m$-dimensional rectifiable set $E$ for which $\mu ({\mathbb R}^n\setminus E)=0$. The  complementary notion of a measure $\mu$ being purely $m$-unrectifiable is  defined by requiring that $\mu$ is singular with  respect to all $m$-rectifiable  measures (cf. also [[Mutually-singular measures|Mutually-singular  measures]]). Thus, in particular, a Borel set $E$ is $m$-rectifiable if  and only if the measure $\mu$ defined by $\mu (A) := \mathcal{H}^m (A\cap E)$ (i.e. the restriction  of $\mathcal{H}^m$ to $E$ is $m$-rectifiable); this  allows one to study rectifiable sets through $m$-rectifiable measures.
−
\[
 
−
T:BV (\Omega)\to L^1 (\partial \Omega, \mathcal{H}^{n-1})
 
−
\]
 
−
such that the following identity holds for any test field $\varphi\in C^\infty (\mathbb R^n,\mathbb R^n)$:
 
−
\[
 
−
\int_\Omega f (x)\, {\rm div} \, \varphi (x)\, dx = -\int_\Omega \varphi (x)\cdot d\mu (x) + \int_{\partial \Omega}
 
−
(\varphi (x)\cdot \nu (x))\, Tf (x)\, d\mathcal{H}^{n-1} (x)
 
−
\]
 
−
(where $\nu$ denotes the exterior unit normal to $\partial \Omega$). In particular, if $f\in C^1 (\overline{\Omega})$, then $Tf$ is simply the restriction of $f$ to $\partial \Omega$.
 
  
−
The theorem holds also for Lipschitz domains (cp. with Theorem 1 of Section 5.3 in {{Cite|EG}}). By a Theorem of Gagliardo, see {{Cite|Ga}}, the trace operator is in fact onto, even when restricted to $W^{1,1} (\Omega)$.
+
====Besicovitch-Marstrand-Preiss theorem====
 +
It is common in analysis to  construct measures as  solutions to equations, and one would like to be  able to deduce  something about the structure of these measures (for example, that they  are rectifiable). Often, the only a priori  information available is  some limited metric information about the  measure, perhaps how the mass  of small balls grows with radius (cp. with [[Density of a set]]). Probably  the strongest known result in this direction is Preiss'  density theorem, which generalizes earlier results of Besicovitch and  Marstrand. The following theorem summarizes both the deep results of Marstrand and Preiss (cp. with {{Cite|De}}).  
  
−
===Pointwise properties===
+
'''Theorem 9'''
−
In this section we fix an open set $\Omega\subset \mathbb R^n$ with $n\geq 2$ and let $u\in BV (\Omega)$ be any given function. The proofs of all claims can be found in Section 3.7 of {{Cite|AFP}} or in Section 5.9 of {{Cite|EG}}
+
Let  $\mu$ be a locally finite Radon measure in the euclidean space $\mathbb R^n$ and $\alpha$ a nonnegative real number. Then the $\alpha$-dimensional density
−
====Approximate continuity====
 
−
There is a Borel set $S_u$ with $\sigma$-finite [[Hausdorff measure|$\mathcal{H}^{n-1}$ measure]] such that $u$ the [[Approximate limit|approximate limit]] of $u$ exists at ''every'' $x\not\in S_u$.
 
−
====Jump set====
 
−
There is a set $J_u\subset S_u$ such that $\mathcal{H}^{n-1} (S_u\setminus J_u)$ and where ''approximate right and left limits'' exist everywhere in the following sense. If $x\in J_u$, then there is a unit vector $\nu_x$ and two values $u^+ (x),\, u^- (x)\in\mathbb R$ such that, if we denote with $B^\pm$ the half balls
 
−
\[
 
−
B^+ =\{y: |y|<1 \quad\mbox{and}\quad (y-x)\cdot \nu_x > 0\}\qquad B^- = \{y: |y|<1 \quad\mbox{and}\quad(y-x)\cdot \nu_x < 0\}\, ,
 
−
\]
 
−
then
 
 
\[
 
\[
−
u^+ (x) = {\rm ap} \lim_{y\in B^+, y \to x} u(y)
+
\lim_{r\downarrow 0} \frac{\mu (B_r (x))}{r^\alpha}
 
\]
 
\]
 +
exists, it is  finite and  positive
 +
at $\mu$-a.e. $x\in \mathbb R^n$ if and  only  $\alpha$ is an integer $k$ and there are a rectifiable  $k$-dimensional  Borel set $E$ and a Borel function $f: E\to ]0,  \infty[$ such that
 
\[
 
\[
−
u^- (x) = {\rm ap} \lim_{y\in B^-, y \to x} u(y)
+
\mu (A) = \int_{A\cap E} f\, d\mathcal{H}^k\qquad \mbox{for any Borel } A\subset \mathbb R^n\, .
 
\]
 
\]
−
(for the definition of ${\rm ap}\lim$ see [[Approximate limit]]).
 
−
====Precise representative====
 
−
Using the properties above it is possible to assign a value to $u$ at every point $x\not \in (S_u\setminus J_u)$. Namely,
 
  
−
'''Definition 17'''
+
====Tangent measures====
−
The precise representative of $u\in BV (\Omega)$ is the Borel measurable function defined by
+
Preiss' main tool in proving that the existence of a density implies the rectifiability of the measure was the notion of tangent measure. A non-zero Radon measure $\nu$ is a tangent  measure of $\mu$ at $x$ if there are  sequences $r_i\downarrow 0$ and $c_i$ such that, for all  continuous real-valued functions $\phi$ with compact support,
 
\[
 
\[
−
\tilde{u} (x) =\left\{
+
\lim_{i\to\infty} c_i \int \phi \left(\frac{y-x}{r_i}\right)\, d\mu (y) = \int \phi (y)\, d\nu (y)\, .
−
\begin{array}{ll}
 
−
{\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if $x\not\in S_u$}\\
 
−
\frac{u^+ (x) + u^- (x)}{2} &\mbox{if $x\in J_u$,}
 
−
\end{array}\right.
 
 
\]
 
\]
−
which coincides with $u$ $\lambda$-a.e..
+
Thus, an  $m$-rectifiable measure will, for almost-every point, have tangent measures which are  multiples of $m$-dimensional  Hausdorff measure restricted to the approximate tangent plane at that  point (cp. with [[Rectifiable set]]); for unrectifiable measures, the set of tangent measures will  usually be much richer
−
====Rectifiability of the jump set====
+
(see {{Cite|ON}}). The utility of the notion lies in the fact that  tangent measures often possess more regularity than the original  measure, thus allowing a wider range of analytical techniques to be used  upon them.
−
The set $J_u$ is [[Rectifiable set|rectifiable]], i.e. up to a set of $\mathcal{H}^{n-1}$-measure zero it can be covered with countably many $C^1$ hypersurfaces. Moreover, at $\mathcal{H}^{n-1}$-a.e. $x\in J_u$ the vector $\nu (x)$ is orthogonal to the approximate tangent space to $J_u$ at $x$ (see [[Rectifiable set]] for the relevant definitions).
 
−
The vector $\nu (x)$ can be chosen so that $x\mapsto \nu (x)$ is a [[Borel function]].
 
−
====Approximate differentiability====
 
−
$u$ is [[Approximate differentiability|approximately differentiable]] at $\lambda$-a.e. $x\in \Omega$. We denote by $\nabla u (x)$ the vector of approximate partial derivaties of $u$ at $x$ (see [[Approximate differentiability]] for the relevant definition). The map $x\mapsto \nabla u (x)$ is [[Measurable function|Lebesgue measurable]].
 
−
====Structure theorem====
 
−
It is possible to relate the pointwise properties of $u$ with the measure-theoretic properties of the generalized derivative $Du$. In this way we gain a suitable generalization of the [[Lebesgue decomposition]] (however this generalization holds ''only'' at the level of the generalized derivative). More precisely we have the following
 
  
−
'''Theorem 18'''
+
=Caccioppoli sets=
−
According to the [[Radon-Nikodym theorem]] $Du$ can be decomposed as $Du^a + Du^s$, where $Du^a$ is absolutely continuous with respect the Lebesgue measure $\lambda$ and $Du^s$ is singular. We then have $Du^a = \nabla u\, \lambda$. Moreover, the measure $Du^s$ can be decomposed as $Du^c+ Du^j$ (called, respectively, Cantor part and Jump part of $Du$) where
 
−
* $Du^c (E) =0$ for every Borel set with $\mathcal{H}^{n-1} (E) <\infty$;
 
−
* For any Borel set $E$ we have the identity
 
−
\begin{equation}\label{e:structure}
 
−
Du^j (E) = \int_{E\cap J_u} (u^+ (x)-u^-(x))\, \nu (x)\, d\mathcal{H}^{n-1} (x)\, .
 
−
\end{equation}
 
−
====Vector-valued case====
 
−
All the properties listed in the previous sections hold for vector-valued functions $u\in BV (\Omega, \mathbb R^n)$. In \eqref{e:structure} we just need to replace
 
−
* $\nabla (x)$ with the [[Jacobian|Jacobi matrix]], whose entries are the approximate partial derivatives of the single coordinate functions,
 
−
* $(u^+ (x)-u^- (x))\,\nu (x)$ with $(u^+ (x)-u^- (x))\otimes \nu (x)$.
 
  
−
==Slicing==
+
A possible (and quite common) definition of perimeter of a measurable set $E\subset \mathbb R^n$ is
−
The restrictions of a $BV$ function on the lines parallel to a given direction are themselves functions of bounded variation ''almost always''. More precisely, given a set $\Omega\subset \mathbb R^m$, a measurable function $u:\Omega\to\mathbb R$, a direction $\nu\in \mathbb S^{n-1}$ and the subspace $\pi$ perpendicular to $\nu$, for every $x\in \pi$ we set
 
 
\[
 
\[
−
\Omega_x:=\{t\in\mathbb R: x+t\nu\in\Omega\}
+
{\rm Per} (E) := \inf    \left\{ \liminf_k\; \mathcal{H}^{n-1} (\partial E_k):\;\{E_k\} \mbox{    is a sequence of smooth sets with } \lambda (E\bigtriangleup E_k) \to    0\right\}\, ,
 
\]
 
\]
−
and we define the sections $u_x:\Omega_x\to\mathbb R$ as $u_x (t):= u (x+t\nu)$. We then have
+
where $\lambda$ denotes the Lebesgue measure. Measurable sets $E$ such that ${\rm Per} (E) < \infty$ are called ''sets of finite perimeter'' or ''Caccioppoli sets''. A localized notion is also possible: if $\Omega$ is an open set, it customary to define ${\rm Per} (E, \Omega)$ by replacing $\mathcal{H}^{n-1} (\partial E_k)$ with  $\mathcal{H}^{n-1} ((\partial E_k)\cap \Omega)$ in the formula above.
−
 
 
−
'''Theorem 19'''
 
−
If $\Omega$ is an open set, $u\in BV (\Omega)$ and $\nu\in\mathbb S^{n-1}$, then $u_x\in BV (\Omega_x)$ for a.e. $x\in\pi$ (with respect to the $n-1$ dimensional measure) and
 
−
\begin{equation}\label{e:slicing}
 
−
\int_\pi \|u_x\|_{BV (\Omega_x)}\, dx\leq \|u\|_{BV (\Omega)}\, .
 
−
\end{equation}
 
−
Viceversa, if $u\in L^1 (\Omega)$ and there are $n$ linearly independent directions $\nu_1, \ldots, \nu_n$ such that $u_x\in BV (\Omega_x)$ for a.e. $x\in\pi_i$ and the corresponding integrals in \eqref{e:slicing} are finite, then $u\in BV (\Omega)$.
 
−
 
 
−
For a proof see Section 5.10 in {{Cite|EG}} or Section 3.11 in {{Cite|EG}}.
 
−
===Tonelli-Cesari variation===
 
−
Combining '''Theorem 9''' with '''Theorem 19''' we then conclude that, if $u\in BV (\Omega)$ and $\nu\in\mathbb S^{n-1}$, then for a.e. $x$ there is a function $\widetilde{u_x}$ which coincides with $u_x$ for $\lambda$-a.e. $t$ and such that the classical total variation (in the sense of '''Definition 1''') of $\widetilde{u_x}$ is finite. However, more can be proved, i.e. a.e. section of the ''precise representative'' of $u$ has bounded variation in the classical sense
 
−
 
 
−
'''Theorem 20'''
 
−
Let $u\in BV (\Omega)$ and $\tilde{u}$ the precise representative of $u$ defined in '''Definition 17'''. For every direction $\nu\in\mathbb S^{n-1}$ and a.e. $x$ in the perpendicular vector subspace $\pi$ the section $\tilde{u}_x$ has bounded total variation in the sense of '''Definition 1'''.
 
  
−
For the proof, see Theorem 3.107 of {{Cite|AFP}. '''Theorem 20''' shows that the modern definition of a $BV (\mathbb R^2)$ function coincides with the one proposed by Cesari in {{Cite|Ce}} as a modification of [[Tonelli plane variation|Tonelli's plabe variation]]. More precisely
+
This definition is in  the spirit of the original work of Caccioppoli where the  approximating sets instead of being  smooth were required to be  polytopes (cp. with {{Cite|Ca}}). It was a  fundamental discovery of De  Giorgi that Caccioppoli's Perimeter has  indeed both a functional and measure-theoretic  interpretation: the functional interpretation, given below, is indeed taken as definition by most authors, whereas the above characterization of the perimeter is then conclude. The theory of Caccioppoli set was first set forth by De Giorgi to solve the Plateau's problem in codimension 1 and study the isoperimetric problem.  
  
−
'''Definition 21'''
+
==Functions of bounded variation==
−
Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as
+
If $E$ is a measurable set, then $E$ is a Caccioppoli set if and only if the indicator function
 
\[
 
\[
−
V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\,
+
{\bf 1}_E (x):= \left\{
−
\]
+
\begin{array}{ll}
−
and the Tonelli-Cesari variation as
+
1 \quad \mbox{if } x\in E\\
−
\[
+
0 \quad \mbox{otherwise}
−
V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\mbox{$\lambda$-a.e.}\right\}\, .
+
\end{array}
 +
\right.
 
\]
 
\]
−
 
+
has finite [[Variation of a function|variation]] $V ({\bf 1}_E, \Omega)$. It then turns out that
−
'''Corollary 22'''
 
−
If $f\in L^1 (\mathbb R^2)$, then $V (f, \mathbb R^2)<\infty$ if and only if $V_{TC} (f)<\infty$.
 
−
 
 
−
Indeed it is possible to show that $V (f,\mathbb R^2)\leq V_{TC} (f) \leq \sqrt{2} V (f,\mathbb R^2)$.
 
−
 
 
−
==Caccioppoli sets==
 
−
A special class of $BV$ functions which play a fundamental role in the theory (and had also a pivotal role in its historical development) is the set of those $f\in BV$ which takes only the values $0$ and $1$ and are, therefore, the indicator functions of a set.
 
−
 
 
−
'''Definition 23'''
 
−
Let $\Omega\subset \mathbb R^n$ be an open set and $E\subset \Omega$ a measurable set such that ${\bf 1}_E\in BV (\Omega)$. The $E$ is called a ''Caccioppoli set'' or a ''set of finite perimeter'' and its perimeter in $\Omega$ is defined to be
 
 
\[
 
\[
 
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 
\]
 
\]
 +
If in addition the Lebesgue measure of $E$ is finite, ${\bf 1}_E$ is a [[Function of bounded variation|function of bounded variation]].
  
−
A primary example is given by those open sets $E\subset \Omega$ which have a $C^1$ topological boundary $\partial E$ such that $\mathcal{H}^{n-1} ((\partial E)\cap\Omega) < \infty)$. If we denote by $\nu$ the exterior unit normal field at $\partial E$, the [[Divergence|divergence theorem]] we then have
+
If $E\subset \Omega$ has a $C^1$ topological boundary $\partial E$ with $\mathcal{H}^{n-1}   ((\partial E)\cap\Omega) < \infty$, then it is a Caccioppoli set and if we denote by $\nu$ the   exterior unit normal field at $\partial E$, the [[Divergence|divergence   theorem]]  
 +
implies
 
\begin{equation}\label{e:divergenza1}
 
\begin{equation}\label{e:divergenza1}
−
\int {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\, \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\, d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\, .
+
\int   {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\,   \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\,   d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\,   .
 
\end{equation}
 
\end{equation}
−
It turns then out by that $V ({\bf 1}_E, \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$, see '''Definition 13''', and hence ${\bf 1}_E\in BV_{loc} (\Omega)$ (for having ${\bf 1}_E\in BV (\Omega)$ we need the additional condition $\lambda (E)<\infty$). In particular, if
+
Thus $V ({\bf 1}_E, \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$ and hence ${\bf 1}_E\in BV_{loc} (\Omega)$ (for having ${\bf 1}_E\in BV (\Omega)$ we need the additional condition $\lambda (E)<\infty$). In particular, if
 
we introduce the vector measure
 
we introduce the vector measure
 
\[
 
\[
 
\mu (A) := - \int_{A\cap E} \nu\, d\mathcal{H}^{n-1}\, ,
 
\mu (A) := - \int_{A\cap E} \nu\, d\mathcal{H}^{n-1}\, ,
 
\]
 
\]
−
\eqref{e:divergenza1} is then simply the identity $D{\bf 1}_E =\mu$.
+
\eqref{e:divergenza1} is then simply the identity $D{\bf 1}_E =\mu$ in the sense of distributions.
  
−
A possible (and quite common) alternative definition of perimeter is
+
For general Caccioppoli sets, it is possible to identify an appropriate notion of ''measure-theoretic boundary'', which is rectifiable and whose Hausdorff measure coincides with the perimeter. A corresponding generalization of the divergence theorem holds. See the section '''Reduced boundary and structure theorem''' of [[Function of bounded variation]] for more details.
−
\[
+
==Plateau's problem in codimension 1==
−
\inf \left\{ \liminf_n\; \mathcal{H}^{n-1} (\partial E_k):\;\mbox{$\{E_k\}$ is a sequence of smooth sets with $\lambda (E\bigtriangleup E_k) \to 0$}\right\}\, .
+
The Caccioppoli sets were first used by De Giorgi to formulate the Plateau's problem in codimension $1$ in the following fashion. Consider two smooth open sets $\Omega, U \subset \mathbb R^n$ such that $\partial U$ and $\partial \Omega$ intersect transversally in a smooth $(n-2)$-dimensional submanifold $\Sigma$.  
−
\]
 
−
This is in the spirit of the original definition of Caccioppoli where the approximating sets instead of being smooth were required to be polytopes (cp. with {{Cite|Ca}}). It was a fundamental discovery of De Giorgi that Caccioppoli's Perimeter has indeed both a functional (as above) and measure-theoretic (see below) interpretation.
 
  
−
===Characterization through density===
+
'''Definition 10'''
−
The following structure theorem, first proved by De Giorgi in his pioneering works, gives a quite precise description of the [[Density of a set|Lebesgue density]] of a generic Caccioppoli set $E$ at most point $x$. Recall that such density is defined as
+
A genealized solution of the Plateau problem in $U$ relative to the boundary $\Sigma$ is a Caccioppoli set $E\subset \mathbb R^n$ such that $E\setminus U = \Omega\setminus U$ and has least perimeter among all such sets.
−
\begin{equation}\label{e:density}
 
−
\theta^n (E,x) =\lim_{r\downarrow 0} \frac{\lambda (E\cap B_r (x))}{\lambda (B_r (x))}\, ,
 
−
\end{equation}
 
−
provided the limit exists.
 
  
−
'''Theorem 24'''
+
Standard compactness properties of the space of functions of bounded variation leads then to the following fundamental existence theorem.
−
If $E\subset\Omega$ is a Caccioppoli set then the limit on the right hand side of \eqref{e:density} exists and takes one of the values $\{0,\frac{1}{2}, 1\}$ for $\mathcal{H}^{n-1}$-a.e. $x$. Moreover the set of points where the density is neither one nor zero or does not exist has finite $\mathcal{H}^{n-1}$ measure. This set is called ''essential boundary'' and denoted by $\partial^* E$ by some authors (see {{Cite|AFP}}) and by $\partial_* E$ by others (see {{Cite|EG}}).
 
  
−
See Theorem 3.61 in {{Cite|AFP}}. In what follows we will stcik to the notation of {{Cite|AFP}} and use $\partial^* E$ for the essential boundary. The converse of '''Theorem 24''' is also true and it is a deep theorem by Federer: see Section 5.11 of {{Cite|EG}}.
+
'''Theorem 11'''
−
===Reduced boundary and structure theorem===
+
There is a generalized solution as defined above, namely the infimum of ${\rm Per}\, (E)$ among all Caccioppoli sets $E$ with $E\setminus U = \Omega\setminus U$ is attained by some set $F$.
−
The essential boundary of a Caccioppoli set can be analyzed further.
 
  
−
'''Definition 25'''
+
==Regularity theory==
−
If $E\subset\Omega$ is a Caccioppoli set the ''reduced boundary'' of $E$ is defined as
+
===Bernstein's problem===
−
\[
+
====Simons' cone====
−
\mathcal{F}  E := \left\{ x\in\Omega : \nu_E (x) := \lim_{r\downarrow 0} \frac{D{\bf  1}_E (B_r(x))}{|D {\bf 1}_E| (B_r(x))}\;\; \mbox{exists and $|\nu_E  (x)|=1$}\right\}\, .
+
===De Giorgi's $\varepsilon$-regularity theorem===
−
\]
+
===Simons' inequality and solution of the Bernstein's problem===
−
$\nu_E$ is called the ''measure theoretic'' inner normal.
+
====Stable surfaces====
 +
===Federer's estimate of the singular set===
 +
===Simon's rectifiability theorem===
 +
==Mumford Shah conjecture==
  
−
We then have the following fundamental result, due to De Giorgi (for a proof see Section 3.5 of {{Cite|AFP}}).
+
=Currents=
 +
==Federer-Fleming theory==
 +
===Compactness for integral currents===
 +
===Deformation theorem===
 +
==Plateau's problem in any codimension==
 +
==Regularity theory==
 +
===Almgren's $\varepsilon$-regularity theorem===
 +
===Almgren's big regularity paper===
 +
==Currents in metric spaces==
 +
=Varifolds=
 +
==General theory==
 +
==Rectifiable and integral varifolds==
 +
==Regularity theory==
 +
===Allard's rectifiability theorem===
 +
===Allard's $\varepsilon$-regularity theorem===
 +
=Calculus of variations in the large=
 +
==Pitts' theory==
 +
===Schoen-Simon curvature estimates===
 +
===The Willmore conjecture===
 +
==Smith's theorem and generalizations==
 +
===Applications to topology===
 +
=Uniqueness of tangent cones=
 +
==White's theorem==
 +
==Simon's theorem==
 +
===Lojasievicz inequality===
 +
=Soap films=
 +
==Almgren's $\varepsilon-\delta$ minimal sets==
 +
==Taylor's theorem==
 +
==Double-bubble conjecture==
 +
=Notable applications=
  
−
'''Theorem 26'''
 
−
For  any $x\in \mathcal{F} E$ the Lebesgue density $\theta^n (E,x)$ is equal  to $\frac{1}{2}$ and hence the reduced boundary is a subset of the  essential boundary (and, by Theorem 23, $\mathcal{H}^{n-1} (\partial^*  E\setminus\mathcal{F} E) = 0$). The set $\mathcal{F} E$ is a rectifiable  set and $\nu_E$ is orthogonal to it $\mathcal{H}^{n-1}$-a.e.. Finally  we have the identity
 
−
\begin{equation}\label{e:structure2}
 
−
D {\bf 1}_E (A) = \int_{A\cap \mathcal{F} E} \nu_E (x)\, d\mathcal{H}^{n-1} (x)\, .
 
−
\end{equation}
 
−
===Generalized divergence theorem===
 
−
'''Theorem 26''' can also be interpreted as a far-reaching generalization of the divergence theorem. We have namely
 
  
−
'''Corollary 27'''
+
=References=
−
Assume  that $E\subset \Omega$ is a Caccioppoli set, $\mathcal{F} E$ its  reduced boundary and $\nu_E$ its measure theoretic inner normal. Then
 
−
\begin{equation}\label{e:div_thm}
 
−
\int_E  {\rm div}\, \varphi\, d\lambda = \int_{\mathcal{F} E} \nu_E \cdot  \varphi\, d\mathcal{H}^{n-1} \qquad \forall \varphi\in C^1_c (\Omega,  \mathbb R^n)\, .
 
−
\end{equation}
 
−
 
 
−
Oberve  therefore that  $\mathcal{F} E$ is, from the point of view of the  divergence theorem,  the ''correct notion'' of boundary. It is not  difficult to give examples  of Caccioppoli open sets with topological  boundary which has positive  Lebesgue measure: for these sets  $\mathcal{F} E$ is indeed a very thin  portion of the topological  boundary!
 
−
===Isoperimetric inequality===
 
−
The classical [[Isoperimetric inequality, classical|isoperimetric inequality]] can be generalized also to Caccioppoli sets. In particular the following fundamental result was first proved by De Giorgi, see {{Cite|DG2}}.
 
−
 
 
−
'''Theorem 28'''
 
−
Let $\alpha (n):=\frac{n}{n-1}$, denote by $B_1$ the unit ball of $\mathbb R^n$ centered at the origin and set
 
−
\[
 
−
C(n):=\frac{\lambda (B_1)}{(\mathcal{H}^{n-1} (\partial B_1))^\alpha}\, .
 
−
\]
 
−
Then
 
−
\[
 
−
\lambda (E) \leq C \Big({\rm Per}\, (E, \mathbb R^n)\Big)^{\alpha}\, .
 
−
\qquad \mbox{for any Caccioppoli set $E\subset\mathbb R^n$}.
 
−
\]
 
−
 
 
−
A ''relative isoperimetric'' inequality holds also in extension domains $\Omega$, see Exercise 3.13 of {{Cite|AFP}}.
 
−
===Coarea formula===
 
−
An important tool which allows often to reduce problems for $BV$ functions to problems for Caccioppoli sets is the following generalization of the [[Coarea formula]], first proved by Fleming and Rishel in {{Cite|FR}}.
 
−
 
 
−
'''Theorem 29'''
 
−
For any open set $\Omega\subset \mathbb R^n$ and any $u\in L^1 (\Omega)$, the map $t\mapsto {\rm Per}\, (\{u>t\}, \Omega)$ is Lebesgue measurable and one has
 
−
\[
 
−
V (u,\Omega) = \int_{-\infty}^\infty {\rm Per}\, (\{u>t\}, \Omega)\, dt\,
 
−
\]
 
−
In  particular, if $u\in BV (\Omega)$, then $U_t:=\{u>t\}$ is a Caccioppoli  set for a.e. $t$ and, for any Borel set  $B\subset \Omega$,
 
−
\[
 
−
|Du| (B) = \int_{-\infty}^\infty |D{\bf 1}_{U_t}| (B)\, dt \qquad\mbox{and}\qquad
 
−
Du (B) = \int_{-\infty}^\infty D{\bf 1}_{U_t} (B)\, dt\,
 
−
\]
 
−
(where the maps $t\mapsto |D{\bf 1}_{U_t}| (B)$ and $t\mapsto D{\bf 1}_{U_t} (B)$ are both Lebesgue measurable).
 
−
 
 
−
Cp. with Theorem 3.40 in {{Cite|AFP}}. In fact the proofs of the Structure Theorem 17 and of the fine pointwise properties of $BV$ functions rely heavily upon the coarea formula and the structure theorem for Caccioppoli sets.
 
−
==Volpert chain rule==
 
−
If $\Omega$ is a bounded open set, $u\in BV (\Omega)$ and $\varphi$ is a Lipschitz function of one real variable, it is relatively easy to show that $\varphi\circ u$ is a $BV$ function and that $V (\varphi\circ u)\leq {\rm Lip}\, \varphi\, V (u, \Omega)$, where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$. Indeed this assertion is a simple corollary of '''Theorem 15''' (cp. with the proof of Theorem 3.96 in {{Cite|AFP}}). A theorem due to Volpert (see {{Cite|Vo}}) gives also, for $\varphi\in C^1$ a description of $D (\varphi\circ u)$ in terms of $Du$ and $\varphi'$. More precisely
 
−
 
 
−
'''Theorem 30'''
 
−
Let $\Omega$ be a bounded open set, $u\in BV (\Omega)$ and $\varphi\in C^1 (\mathbb R)$ a Lipschitz function. If
 
−
* $\tilde{u}$ denotes the precise representative of $u$ (cp. with '''Definition 17'''),
 
−
* $Du^a$ and $Du^c$ denote the absolutely continuous and Cantor part of $Du$ (cp. with '''Theorem 18'''),
 
−
* $J_u$ denotes the jump set of $u$, $\nu$ a Borel normal vector field to $J_u$ and $u^+$ and $u^-$ the approximate left and right limit (cp. with '''Definition''' 17)
 
−
then, for any borel set $B\subset\Omega$,
 
−
\[
 
−
D (\varphi \circ u) = \int_B \varphi' (u (x))\, d Du^a (x) + \int_B \varphi' (\tilde{u} (x)) \, d Du^c (x)
 
−
+ \int_{J_u\cap B} (\varphi (u^+ (x)) - \varphi (u^-(x)))\, \nu (x)\, d\mathcal{H}^{n-1} (x)\, .
 
−
\]
 
−
 
 
−
Indeed the theorem holds even if $\varphi$ and $u$ are vector-valued (see Theorem 3.96 of {{Cite|AFP}}). The chain rule of Volpert has been generalized by Ambrosio and Dal Maso to Lipschitz $\varphi$ (see {{Cite|AD}}).
 
−
==Alberti's rank-one theorem==
 
−
Consider a map $u\in BV (\Omega, \mathbb R^m)$ and let $Du^j$ be the jump part of $Du$ (cp. with '''Theorem 18'''). The structure theorem implies that
 
−
\[
 
−
Du^j (B) = \int_{J_u \cap B} (f(u^+)-f(u)^-)\otimes\nu (x)\, d\mathcal{H}^{n-1} (x)\, .
 
−
\]
 
−
In other words, if we denote by $\mu$ the measure $\mu (B):= \mathcal{H}^{n-1} (J_u\cap B)$, then $Du^j = M \mu$, where $M$ is Borel map taking values in the cone of rank-one matrices. A deep theorem of Alberti ({{Cite|Al}}) shows that also the Cantor part $Du^c$ has this property.
 
−
 
 
−
'''Theorem 31'''
 
−
If $u\in BV (\Omega, \mathbb R^m)$ then $Du^c = M |Du^c|$, where $M$ is a Borel map taking values in the cone of rank-one matrices (and $|Du^c|$ is the total variation measure of $Du^c$).
 
−
 
 
−
For a readable account of Alberti's original proof see {{Cite|DL}}.
 
−
 
 
−
==Special Functions of bounded variation==
 
−
In {{Cite|DA}}, in order to study variational problems involving free discontinuity (most notably the [[Mumford-Shah functional]]) De Giorgi and Ambrosio considered a closed subspace of the space $BV (\Omega)$ consisting of those elements $u$ for which $Du^c=0$ (cp with '''Theorem 18''').
 
−
They called them ''special functions of bounded variations'' and denoted the corresponding space by $SBV (\Omega)$ Though this space is not closed in the weak$^*$ topology, the authors discovered that it still has a useful closure property, suitable for the application to many variational problems. The following, which is a corollary of a more general closure theorem (cp. with Theorem 4.7 in {{Cite|AFP}}), makes clear why, for instance, the space $SBV$ is suitable for a flexible existence of minimizers of the Mumford-Shah energy.
 
−
 
 
−
'''Theorem 32'''
 
−
Let $\{u_h\}\subset SBV (\Omega)$ be a sequence such that
 
−
* $\mathcal{H}^{n-1} (J_{u_h})$ is bounded by a constant independent of $h$;
 
−
* there is an increasing function $\varphi\in C (\mathbb R)$ such that $\lim_{t\to\infty} \frac{\varphi (t)}{t} =\infty$ and
 
−
\begin{equation}\label{e:superlinear}
 
−
\limsup_{h\to\infty} \int \varphi (\nabla u (x))\, dx <\infty \, .
 
−
\end{equation}
 
−
* $\|u_h-u\|_{L^1}\to 0$.
 
−
Then the function $u$ belongs also to $SBV (\Omega)$ and, moreover, $Du_h^a\rightharpoonup^\star Du^a$ and $Du_h^j\rightharpoonup^\star Du^j$.
 
−
 
 
−
We refer to Chapter 4 of {{Cite|AFP}} for a comprehensive account of the theory of special functions of bounded variation.
 
−
 
 
−
==Notable applications==
 
−
===Plateau's problem===
 
−
Since their inroduction by De Giorgi, sets of finite perimeter have been successfully employed to prove the existence of hypersurfaces $\Sigma$ minimizing the area among the ones with a fixed given boundary $\Gamma$ (see [[Plateau problem]]). Through the work of several mathematicians (De Giorgi, Fleming, Federer, Almgren and Simons) this lead to the proof that such surface exists in the smooth category in $\mathbb R^n$ for $n\leq 7$ and that the singularities have a rather small dimension for $n\geq 8$. We refer to the book of Giusti {{Cite|Gi}} for a quite thorough account.
 
−
====Isoperimetry====
 
−
Sets of finite perimeter provide also a very natural framework for constrained variational problems such as minimizing the perimeter when the volume of the set is assigned. 
 
−
===Hyperbolic conservation laws===
 
−
The space of $BV$ functions play a fundamental role in the existence of solutions for hyperbolic systems of conservation laws in one space dimension and for scalar conservation laws in several space dimensions. We refer the reader to the textbooks {{Cite|Br}}, {{Cite|Da}} and {{Cite|Se}}.
 
−
===Mumford shah functional===
 
−
The space $SBV (\Omega)$ has been introduced by Ambrosio and De Giorgi to give a suitable space where the existence of minimizers of the Mumford-Shah functional can be approached with the [[Variational calculus|direct methods]] of the calculus of variations.
 
−
===Cahn-Hilliard===
 
−
The [[Cahn-Hilliard equation|Cahn-Hilliard equations]] are elliptic partial differential equations arising in mathematical physics taking the form $\varepsilon^2 \Delta u = f(u)$. They are therefore the Euler Lagrange equation of the energy functional
 
−
\[
 
−
W_\varepsilon (u) := \int_\Omega \left(\varepsilon |\nabla u|^2 + \frac{W(u)}{\varepsilon}\right)\, .
 
−
\]
 
−
These functionals converge, formally, to the area functional as $\varepsilon\downarrow 0$. One way to give a rigorous mathematical account of this assertion is to use the space of Caccioppoli sets and the theory of [[Gamma-convergence]], see for instance {{Cite|DM}}.
 
−
==References==
 
 
{|
 
{|
 
|-
 
|-
−
|valign="top"|{{Ref|Al}}|| G. Alberti, "Rank-one properties for derivatives of functions of bounded variation", Proc. Roy Soc. Edinburgh Sect. A, '''123''' (1993) pp. 239-274
+
|valign="top"|{{Ref|Be1}}|| A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (I). Math. Ann. Vol. 98 (1927), pp. 422-464.
−
|-
 
−
|valign="top"|{{Ref|Am}}|| L. Ambrosio, "Metric space valued functions with bounded variation", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), '''17''' (1990) pp. 291-322.
 
−
|-
 
−
|valign="top"|{{Ref|AD}}|| L. Ambrosio, G. Dal Maso, "A general chain rule for distributional derivatives", Proc. Amer. Math. Soc., '''108''' (1990) pp. 691-792.
 
−
|-
 
−
|valign="top"|{{Ref|AFP}}||    L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The    Clarendon Press,  Oxford University Press, New York, 2000.      {{MR|1857292}}{{ZBL|0957.49001}}
 
−
|-
 
−
|valign="top"|{{Ref|Br}}|| A. Bressan, "Hyperbolic systems of conservation laws",  Cambridge University Press, Cambridge, 2002.
 
−
|-
 
−
|valign="top"|{{Ref|Ca}}|| R. Caccioppoli, "Misura e integrazione sugli insiemei dimensionalmente orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11 and 137-146.
 
−
|-
 
−
|valign="top"|{{Ref|Ce}}|| L. Cesari, "Sulle funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp. 299-313.
 
−
|-
 
−
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
 
−
|-
 
−
|valign="top"|{{Ref|Co}}|| C. M. Dafermos, "Hyperbolic conservation laws in continuum physics", 2nd edition, Springer Verlag, 2005,
 
−
|-
 
−
|valign="top"|{{Ref|DM}}|| G. Dal Maso, "An introduction to $\Gamma$-convergence", Burkhäuser, 1993.
 
−
|-
 
−
|valign="top"|{{Ref|DG}}|| E. De Giorgi, L. Ambrosio, "Un nuovo funzionale nel calcolo delle variazioni", Att. Acc. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8) Mat. Appl., '''82''' (1988) pp. 199-210.
 
−
|-
 
−
|valign="top"|{{Ref|DG}}|| E. De Giorgi, "Su una teoria generale della misura $n-1$-dimensionale in uno spazio a $r$ dimensioni", Ann. Mat. Pura Appl. (4), '''36''' (1954) pp. 191-213.
 
−
|-
 
−
|valign="top"|{{Ref|DG2}}|| E. De Giorgi, "Sulla proprietà isoperimetrica dell'ipersfera, nella classe degli insiemi aventi frontiera orientata di misura finita", Att. Acc. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, '''8''' (1958) pp. 33-44.
 
−
|-
 
−
|valign="top"|{{Ref|DL}}|| C. De Lellis, "A note on Alberti's rank-one theorem", Transport equations and multi-D hyperbolic conservation laws, 61-74, Lect. Notes Unione Mat. Ital., 5, Springer, Berlin, 2008.
 
 
|-
 
|-
−
|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
+
|valign="top"|{{Ref|Be2}}|| A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (II). Math. Ann. Vol. 115 (1938), pp. 296-329.
 
|-
 
|-
−
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of  functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}}
+
|valign="top"|{{Ref|Be3}}|| A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (III). Math. Ann. Vol. 116 (1939), pp. 349-357.
 
|-
 
|-
−
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure  theory". Volume 153 of Die Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New York Inc., New  York, 1969. {{MR|0257325}} {{ZBL|0874.49001}}
+
|valign="top"|{{Ref|Be4}}|| A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions (I) Proc. Cambridge Phil. Soc. Vol. 41 (1945), pp. 103-110.
 
|-
 
|-
−
|valign="top"|{{Ref|Fi}}|| G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico dell'Università di Trieste, vol. I, 1954.
+
|valign="top"|{{Ref|Be4}}|| A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions (II) Proc. Cambridge Phil. Soc. Vol. 42 (1946), pp. 1-10.
 
|-
 
|-
−
|valign="top"|{{Ref|FR}}|| W. H. Fleming, R. Rishel, "An integral formula for total gradient variation", Arch. Math., '''11''' (1960) pp. 218-222.
+
|valign="top"|{{Ref|Ca}}|| R. Caccioppoli, "Misura e  integrazione sugli insiemei dimensionalmente  orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11  and 137-146.
 
|-
 
|-
−
|valign="top"|{{Ref|Ga}}|| E. Gagliardo, "Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in piú variabili", Rend. Sem. Mat. Univ. Padova, '''27''' (1957) pp. 284-305.  
+
|valign="top"|{{Ref|De}}|| C. De Lellis, "Rectifiable sets,  densities and tangent measures" Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. {{MR|2388959}}  {{ZBL|1183.28006}}
 
|-
 
|-
−
|valign="top"|{{Ref|Gi}}|| E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
+
|valign="top"|{{Ref|EG}}||   L.C. Evans, R.F.  Gariepy, "Measure theory  and fine properties of  functions" Studies in  Advanced Mathematics. CRC  Press, Boca Raton, FL,    1992. {{MR|1158660}} {{ZBL|0804.2800}}
 
|-
 
|-
−
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
+
|valign="top"|{{Ref|Fa}}|| K. J. Falconer. "The geometry of  fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University  Press, Cambridge, 1986. {{MR|0867284}} {{ZBL|0587.28004}}  
 
|-
 
|-
−
|valign="top"|{{Ref|HS}}||   E. Hewitt,  K.R. Stromberg,  "Real and abs
+
|valign="top"|{{Ref|Fa2}}|| K. J. Falconer. "Fractal Geometry: Mathematical Foundations and Applications". John Wiley & Sons, 2003.
−
tract analysis" , Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
 
 
|-
 
|-
−
|valign="top"|{{Ref|Jo}}|| C. Jordan,  "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230
+
|valign="top"|{{Ref|Fe}}||     H. Federer, "Geometric measure   theory". Volume 153 of Die  Grundlehren  der mathematischen  Wissenschaften. Springer-Verlag New  York Inc., New  York, 1969.   {{MR|0257325}} {{ZBL|0874.49001}}
 
|-
 
|-
−
|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan (1969). {{MR|0151555}} {{ZBL|0197.03501}}
+
|valign="top"|{{Ref|Ha}}|| F. Hausdorff,  "Dimension and äusseres Mass" ''Math. Ann.'' , '''79''' (1918) pp. 157–179  {{MR|1511917}} {{ZBL|}}  
 
|-
 
|-
−
|valign="top"|{{Ref|To}}|| L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp. 315-320.
+
|valign="top"|{{Ref|KP}}|| S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
 
|-
 
|-
−
|valign="top"|{{Ref|Vi}}|| A. Vitali,"Sulle funzioni integrali", Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., '''40''' 1905 pp. 1021-1034.
+
|valign="top"|{{Ref|Ma}}|| P. Mattila, "Geometry of sets  and measures in Euclidean spaces.  Fractals and rectifiability".    Cambridge Studies in Advanced  Mathematics, 44. Cambridge University      Press, Cambridge, 1995.   {{MR|1333890}} {{ZBL|0911.28005}}
 
|-
 
|-
−
|valign="top"|{{Ref|Vo}}|| A. I. Volpert, "Spaces BV and quasi-linear equations", Math. USSR Sb., '''17''' (1967) pp. 225-267.
+
|valign="top"|{{Ref|Mu}}|| M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
 
|-
 
|-
−
|valign="top"|{{Ref|Zy}}|| A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)
+
|valign="top"|{{Ref|Wh}}|| B. White, "A new proof of Federer's structure theorem for $k$-dimensional subsets of $\mathbb R^n$ ''J. Amer. Math. Soc.'' , '''11''' : 3 (1998) pp. 693–701
 
|-
 
|-
 
|}
 
|}

Latest revision as of 14:54, 20 August 2013

2020 Mathematics Subject Classification: Primary: 49Q15 Secondary: 49Q2049Q0528A7532C3058A2558C35 [MSN][ZBL]


An area of analysis concerned with solving geometric problems via measure-theoretic techniques. The canonical motivating physical problem is probably that investigated experimentally by J. Plateau in the nineteenth century: Given a boundary wire, how does one find the (minimal) soap film which spans it? In mathematical terms: Given a boundary curve, find the surface of minimal area spanning it. (Cf. also Plateau problem.) The many different approaches to solving this problem have found utility in most areas of modern mathematics and geometric measure theory is no exception: techniques and ideas from geometric measure theory have been found useful in the study of partial differential equations, the calculus of variations, geometric analysis, harmonic analysis, and fractals.

Many variational problems (cf. also Variational calculus) are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged class. Much of the work in geometric measure theory has been directed towards placing this informal description on a formal footing appropriate for the study of surfaces.

History

Measure theoretic concepts

One of the central issues of geometric measure theory is to define the concepts of volume, area and length in the uttermost generality. The roots of them are obviously in measure theory. In particular, the concept of outer measure in a metric space plays a fundamental role.

Caratheodory construction

The following is a common construction of metric outer measures in metric spaces $(X, d)$.

Definition 1 If $\mathcal{C}$ is a class of subsets of $X$ containing the empty set, $\nu : \mathcal{C}\to [0, \infty]$ a set function with $\nu (\emptyset) =0$ and $\delta\in ]0, \infty]$, then for every $A\subset X$ we define \[ \mu^\delta (A) := \inf \left\{ \sum_i \nu (E_i) : \{E_i\}_{i\in \mathbb N} \subset \mathcal{C} \mbox{ covers } A \mbox{ and } {\rm diam}\, (E_i) \leq \delta\,\right\}\, \] and \[ \mu (A) := \lim_{\delta\downarrow 0}\; \mu^\delta (A)\, . \]

Observe that the latter limit exists because $\mu^\delta (A)$ is a nonincreasing function of $\delta$. This construction is often called Caratheodory construction. See Section 2.1 of [KP] (cp. also with [Fe]).

Theorem 2 Let $\nu$ and $\mu$ be as in Definition 7. Then $\mu$ is a metric outer measure. Thus there is a $\sigma$-algebra $\mathcal{A}$ which contains the Borel sets and such that the restriction of $\mu$ to $\mathcal{A}$ is $\sigma$-additive.

(Cp. with Claims 1,2 and 3 in the proof of Theorem 1 in Section 2.1 of [EG]: although the reference handles the cases of Hausdorff outer measures, the proof extends verbatim to the setting above).

Hausdorff measures

The Caratheodory construction gives several generalizations of the concept of dimension and volume. The most common is due to Hausdorff (cp. with [Ha]).

Definition 3 The Hausdorff outer measure $\mathcal{H}^\alpha$ is given by such $\mu$ as in Definition 1 when we choose $\mathcal{C} = \mathcal{P} (X)$ and $\nu (A) = \omega_\alpha ({\rm diam}\, (A))^\alpha$, where \[ \omega_\alpha = \frac{\pi^{\alpha/2}}{\Gamma \left(\frac{\alpha}{2}+1\right)}\, \] (cp. with Section 2.1 of [EG]).

When $\alpha$ is a (positive) integer $n$, $\omega_n$ equals the Lebesgue measure of the unit ball in $\mathbb R^n$. With this choice the $n$-dimensional Hausdorff outer measure on the euclidean space $\mathbb R^n$ coincides with the Lebesgue measure. It must be noted, however, that some authors prefer to set the constant $\omega_\alpha$ equal to $1$ (see for instance [Ma]).

Hausdorff dimension

The following is a simple consequence of the definition (cp. with Theorem 4.7 of [Ma]).

Theorem 4 For $0\leq s<t<\infty$ and $A\subset X$ we have

  • $\mathcal{H}^s (A) < \infty \Rightarrow \mathcal{H}^t (A) = 0$;
  • $\mathcal{H}^t (A)>0 \Rightarrow \mathcal{H}^s (A) = \infty$.

The Hausdorff dimension ${\rm dim}_H (A)$ of a subset $A\subset X$ is then defined as

Definition 5 \begin{align*} {\rm dim}_H (A) &= \sup \{s: \mathcal{H}^s (A)> 0\} = \sup \{s: \mathcal{H}^s (A) = \infty\}\\ &=\inf \{t: \mathcal{H}^t (A) = 0\} = \inf \{t: \mathcal{H}^t (A) < \infty\}\, . \end{align*}

Other measures related to the volume

The Caratheodory construction can be used to build other generalizations of the concept of volume, such as

  • The spherical $\alpha$-dimensional outer measures on a metric space $(X,d)$, see Section 2.1.2 of [KP].
  • The Gross outer measures, the Caratheodory outer measures, the integral-geometric outer measures (see also Favard measure) and the Gillespie outer measures in $\mathbb R^n$, see Sections 2.1.3-2.1.4-2.1.5 of [KP] (cp. also with 2.10.2-2.10.3-2.10.4 of [Fe]).

All these measures coincide on sufficiently regular set (in particular on rectifiable sets, see below), but they are, in general, different.

Fractals

Fractals have been originally defined by B.B. Mandelbrot as point sets with non-integer Hausdorff dimension, although this name is nowadays also applied to sets of integer dimensions which are purely unrectifiable. Classical examples of fractals are the Cantor set, the Koch curve and the Julia set of an holomorphic function $f$. Fractals are typically self-similar.

Sets with non-integer Hausdorff dimension have been objects of study of geometric measure theory since the pioneering works of Besicovitch and Marstrand. For an account of the modern mathematical theory of fractals we refer the reader to [Fa2].

Differentiation theorem

It is common in geometric measure theory to construct measures as solutions to geometric problems or to partial differential equations. In these cases measures are often considered as a suitable relaxation of $k$-dimensional surfaces. For instance, given a $k$-dimensional surface $\Gamma\subset \mathbb R^n$, one can natural associate to this surface the measure $\mu (A) := \mathcal{H}^k (A\cap \Gamma)$. It is therefore of interest to understand which assumptions guarantee that a measure has a structure as in the latter example. A fundamental tool in this direction, which is used ubiquitously in geometric measure theory, is the following theorem (cp. with Differentiation of measures), usually credited to Besicovitch (see [Be4] and [Be5]) and which gives an explicit characterization of the Radon-Nykodim decomposition for locally finite Radon measures on the euclidean space.

Theorem 6 (cp. with Theorem 2.12 of [Ma] and Theorem 2 in Section 1.6 of [EG]) Let $\mu$ and $\nu$ be two locally finite Radon measures on $\mathbb R^n$. Then,

  • the limit

\[ f(x) := \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} \] exists at $\mu$-a.e. $x$ and defines a $\mu$-measurable map;

  • the set

\begin{equation}\label{e:singular} S:= \left\{ x: \lim_{r\downarrow 0} \frac{\nu (B_r (x))}{\mu (B_r (x))} = \infty\right\} \end{equation} is $\nu$-measurable and a $\mu$-null set;

  • $\nu$ can be decomposed as $\nu_a + \nu_s$, where

\[ \nu_a (E) = \int_E f\, d\mu \] and \[ \nu_s (E) = \nu (S\cap E)\, . \] Moreover, for $\mu$-a.e. $x$ we have: \begin{equation}\label{e:Lebesgue} \lim_{r\downarrow 0} \frac{1}{\mu (B_r (x))} \int_{B_r (x)} |f(y)-f(x)|\, d\mu (y) = 0\qquad \mbox{and}\qquad \lim_{r\downarrow 0} \frac{\nu_s (B_r (x))}{\mu (B_r (x))}= 0\, . \end{equation}

Covering arguments

Theorem 6 does not hold in general metric spaces. It holds provided the metric space satisfies some properties about covering of sets with balls (cp. with Covering theorems (measure theory)). In fact, aside from their links to the differentiation of measures, both the Vitali and Besicovitch covering Lemmas (see again Covering theorems (measure theory)) and similar arguments play a pivotal role in several fundamental results of geometric measure theory.

Rectifiable sets

Also called countably rectifiable sets, rectifiable sets are central objects of study in Geometric measure theory, cp. with Rectifiable set. Rectifiable sets of the euclidean space are fairly close to $C^1$ submanifolds, a sort of measure-theoretic counterpart of the latter. As such they have an integer Hausdorff dimension. In the special case of $1$-dimensional sets of the euclidean space, they were first introduced by Besicovitch, as a suitable generalization of rectifiable curves. In what follows we will use the terminology $m$-dimensional rectifiable set. Some authors prefer the terminology countably $m$-rectifiable set or, briefly, $m$-rectifiable.

Rectifiable subsets of the Euclidean space $\mathbb R^n$ can be defined in several ways, see Rectifiable set. We adopt here the following one

Definition 7 (cp. with Lemma 11.1 of [Si]) A Borel set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if it has Hausdorff dimension $k$ and there is a countable family of $C^1$ $k$-dimensional submanifolds of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.

The assumption that $E$ is a Borel set might be dropped, but in this case the set might not be $\mathcal{H}^k$-measurable (see Rectifiable set). In what follows we will assume that rectifiable sets $E$ are $\mathcal{H}^k$ measurable: $\sigma$-finite $\mathcal{H}^k$-measurable sets can be decomposed into the union of a Borel set and an $\mathcal{H}^k$-null set.

Besicovitch's works

As already mentioned, the theory of rectifiable $1$-dimensional sets was first laid down by Besicovitch in a series of works (see [Be1], [Be2], [Be3]). In these seminal papers Besicovitch considered and solved several questions which became later pivotal for rectifiable sets of higher dimension. He also introduced many useful tools (such as the Besicovitch covering theorem, see Covering theorems (measure theory)) which have been widely used in different contexts.

One-dimensional sets

The theory of one-dimensional rectifiable sets is somewhat special since much stronger theorems can be proved which fail for higher dimensions. Perhaps the most useful one is the following (cp. with Theorem 3.14 of [Fa]):

Theorem 8 A continuum, i.e. a compact connected set, $E\subset\mathbb R^n$ of finite $\mathcal{H}^1$ measure is always rectifiable and arcwise connected. Indeed it is always the image of a rectifiable curve.

We refer to [Fa] for a comprehensive account of the theory of rectifiable one-dimensional sets.

General dimension and codimension

The main importance of the class of rectifiable sets is that it possesses many of the nice properties of the smooth surfaces which one is seeking to generalize. For example, although, in general, classical tangents may not exist, an $m$-rectifiable set will possess a unique approximate tangent at $\mathcal{H}^m$-almost every point (see Rectifiable set).

Besicovitch-Federer projection theorem

Often, one is faced with the task of showing that some set, which is a solution to the problem under investigation, is in fact rectifiable, and hence possesses some smoothness. A major concern in geometric measure theory is finding criteria which guarantee rectifiability, several of these criteria are listed in Rectifiable set. One of the most striking results in this direction is the Besicovitch–Federer projection theorem, which illustrates the stark difference between rectifiable and unrectifiable sets. This theorem characterize purely unrectifiable $m$-dimensional sets as those sets whose projections are $\mathcal{H}^m$-negligible on almost every $m$-dimensional plane (for the precise statement, we refer to Rectifiable set). This deep result was first proved for $1$-dimensional sets in the plane by A.S. Besicovitch, and later extended to higher dimensions by H. Federer. B. White in [Wh] has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version.

Recitifiable measures

It is also possible (and useful) to define a notion of rectifiability for (locally finite) Radon measures: A Radon measure $\mu$ is said to be $m$-rectifiable if it is absolutely continuous (cf. also Absolute continuity) with respect to the $m$-dimensional Hausdorff measure and there is an $m$-dimensional rectifiable set $E$ for which $\mu ({\mathbb R}^n\setminus E)=0$. The complementary notion of a measure $\mu$ being purely $m$-unrectifiable is defined by requiring that $\mu$ is singular with respect to all $m$-rectifiable measures (cf. also Mutually-singular measures). Thus, in particular, a Borel set $E$ is $m$-rectifiable if and only if the measure $\mu$ defined by $\mu (A) := \mathcal{H}^m (A\cap E)$ (i.e. the restriction of $\mathcal{H}^m$ to $E$ is $m$-rectifiable); this allows one to study rectifiable sets through $m$-rectifiable measures.

Besicovitch-Marstrand-Preiss theorem

It is common in analysis to construct measures as solutions to equations, and one would like to be able to deduce something about the structure of these measures (for example, that they are rectifiable). Often, the only a priori information available is some limited metric information about the measure, perhaps how the mass of small balls grows with radius (cp. with Density of a set). Probably the strongest known result in this direction is Preiss' density theorem, which generalizes earlier results of Besicovitch and Marstrand. The following theorem summarizes both the deep results of Marstrand and Preiss (cp. with [De]).

Theorem 9 Let $\mu$ be a locally finite Radon measure in the euclidean space $\mathbb R^n$ and $\alpha$ a nonnegative real number. Then the $\alpha$-dimensional density \[ \lim_{r\downarrow 0} \frac{\mu (B_r (x))}{r^\alpha} \] exists, it is finite and positive at $\mu$-a.e. $x\in \mathbb R^n$ if and only $\alpha$ is an integer $k$ and there are a rectifiable $k$-dimensional Borel set $E$ and a Borel function $f: E\to ]0, \infty[$ such that \[ \mu (A) = \int_{A\cap E} f\, d\mathcal{H}^k\qquad \mbox{for any Borel } A\subset \mathbb R^n\, . \]

Tangent measures

Preiss' main tool in proving that the existence of a density implies the rectifiability of the measure was the notion of tangent measure. A non-zero Radon measure $\nu$ is a tangent measure of $\mu$ at $x$ if there are sequences $r_i\downarrow 0$ and $c_i$ such that, for all continuous real-valued functions $\phi$ with compact support, \[ \lim_{i\to\infty} c_i \int \phi \left(\frac{y-x}{r_i}\right)\, d\mu (y) = \int \phi (y)\, d\nu (y)\, . \] Thus, an $m$-rectifiable measure will, for almost-every point, have tangent measures which are multiples of $m$-dimensional Hausdorff measure restricted to the approximate tangent plane at that point (cp. with Rectifiable set); for unrectifiable measures, the set of tangent measures will usually be much richer (see [ON]). The utility of the notion lies in the fact that tangent measures often possess more regularity than the original measure, thus allowing a wider range of analytical techniques to be used upon them.

Caccioppoli sets

A possible (and quite common) definition of perimeter of a measurable set $E\subset \mathbb R^n$ is \[ {\rm Per} (E) := \inf \left\{ \liminf_k\; \mathcal{H}^{n-1} (\partial E_k):\;\{E_k\} \mbox{ is a sequence of smooth sets with } \lambda (E\bigtriangleup E_k) \to 0\right\}\, , \] where $\lambda$ denotes the Lebesgue measure. Measurable sets $E$ such that ${\rm Per} (E) < \infty$ are called sets of finite perimeter or Caccioppoli sets. A localized notion is also possible: if $\Omega$ is an open set, it customary to define ${\rm Per} (E, \Omega)$ by replacing $\mathcal{H}^{n-1} (\partial E_k)$ with $\mathcal{H}^{n-1} ((\partial E_k)\cap \Omega)$ in the formula above.

This definition is in the spirit of the original work of Caccioppoli where the approximating sets instead of being smooth were required to be polytopes (cp. with [Ca]). It was a fundamental discovery of De Giorgi that Caccioppoli's Perimeter has indeed both a functional and measure-theoretic interpretation: the functional interpretation, given below, is indeed taken as definition by most authors, whereas the above characterization of the perimeter is then conclude. The theory of Caccioppoli set was first set forth by De Giorgi to solve the Plateau's problem in codimension 1 and study the isoperimetric problem.

Functions of bounded variation

If $E$ is a measurable set, then $E$ is a Caccioppoli set if and only if the indicator function \[ {\bf 1}_E (x):= \left\{ \begin{array}{ll} 1 \quad \mbox{if } x\in E\\ 0 \quad \mbox{otherwise} \end{array} \right. \] has finite variation $V ({\bf 1}_E, \Omega)$. It then turns out that \[ {\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, . \] If in addition the Lebesgue measure of $E$ is finite, ${\bf 1}_E$ is a function of bounded variation.

If $E\subset \Omega$ has a $C^1$ topological boundary $\partial E$ with $\mathcal{H}^{n-1} ((\partial E)\cap\Omega) < \infty$, then it is a Caccioppoli set and if we denote by $\nu$ the exterior unit normal field at $\partial E$, the divergence theorem implies \begin{equation}\label{e:divergenza1} \int {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\, \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\, d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\, . \end{equation} Thus $V ({\bf 1}_E, \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$ and hence ${\bf 1}_E\in BV_{loc} (\Omega)$ (for having ${\bf 1}_E\in BV (\Omega)$ we need the additional condition $\lambda (E)<\infty$). In particular, if we introduce the vector measure \[ \mu (A) := - \int_{A\cap E} \nu\, d\mathcal{H}^{n-1}\, , \] \eqref{e:divergenza1} is then simply the identity $D{\bf 1}_E =\mu$ in the sense of distributions.

For general Caccioppoli sets, it is possible to identify an appropriate notion of measure-theoretic boundary, which is rectifiable and whose Hausdorff measure coincides with the perimeter. A corresponding generalization of the divergence theorem holds. See the section Reduced boundary and structure theorem of Function of bounded variation for more details.

Plateau's problem in codimension 1

The Caccioppoli sets were first used by De Giorgi to formulate the Plateau's problem in codimension $1$ in the following fashion. Consider two smooth open sets $\Omega, U \subset \mathbb R^n$ such that $\partial U$ and $\partial \Omega$ intersect transversally in a smooth $(n-2)$-dimensional submanifold $\Sigma$.

Definition 10 A genealized solution of the Plateau problem in $U$ relative to the boundary $\Sigma$ is a Caccioppoli set $E\subset \mathbb R^n$ such that $E\setminus U = \Omega\setminus U$ and has least perimeter among all such sets.

Standard compactness properties of the space of functions of bounded variation leads then to the following fundamental existence theorem.

Theorem 11 There is a generalized solution as defined above, namely the infimum of ${\rm Per}\, (E)$ among all Caccioppoli sets $E$ with $E\setminus U = \Omega\setminus U$ is attained by some set $F$.

Regularity theory

Bernstein's problem

Simons' cone

De Giorgi's $\varepsilon$-regularity theorem

Simons' inequality and solution of the Bernstein's problem

Stable surfaces

Federer's estimate of the singular set

Simon's rectifiability theorem

Mumford Shah conjecture

Currents

Federer-Fleming theory

Compactness for integral currents

Deformation theorem

Plateau's problem in any codimension

Regularity theory

Almgren's $\varepsilon$-regularity theorem

Almgren's big regularity paper

Currents in metric spaces

Varifolds

General theory

Rectifiable and integral varifolds

Regularity theory

Allard's rectifiability theorem

Allard's $\varepsilon$-regularity theorem

Calculus of variations in the large

Pitts' theory

Schoen-Simon curvature estimates

The Willmore conjecture

Smith's theorem and generalizations

Applications to topology

Uniqueness of tangent cones

White's theorem

Simon's theorem

Lojasievicz inequality

Soap films

Almgren's $\varepsilon-\delta$ minimal sets

Taylor's theorem

Double-bubble conjecture

Notable applications

References

[Be1] A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (I). Math. Ann. Vol. 98 (1927), pp. 422-464.
[Be2] A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (II). Math. Ann. Vol. 115 (1938), pp. 296-329.
[Be3] A. S. Besicovitch, On the fundamental properties of linearly measurable plane sets of points (III). Math. Ann. Vol. 116 (1939), pp. 349-357.
[Be4] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions (I) Proc. Cambridge Phil. Soc. Vol. 41 (1945), pp. 103-110.
[Be4] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions (II) Proc. Cambridge Phil. Soc. Vol. 42 (1946), pp. 1-10.
[Ca] R. Caccioppoli, "Misura e integrazione sugli insiemei dimensionalmente orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11 and 137-146.
[De] C. De Lellis, "Rectifiable sets, densities and tangent measures" Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. MR2388959 Zbl 1183.28006
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fa] K. J. Falconer. "The geometry of fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. MR0867284 Zbl 0587.28004
[Fa2] K. J. Falconer. "Fractal Geometry: Mathematical Foundations and Applications". John Wiley & Sons, 2003.
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ha] F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
[KP] S. G. Krantz, H. Parks, "Geometric Integration Theory", Birkhäuser (2008).
[Ma] P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
[Mu] M. E. Munroe, "Introduction to Measure and Integration". Addison Wesley (1953).
[Wh] B. White, "A new proof of Federer's structure theorem for $k$-dimensional subsets of $\mathbb R^n$ J. Amer. Math. Soc. , 11 : 3 (1998) pp. 693–701
How to Cite This Entry:
Camillo.delellis/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Camillo.delellis/sandbox&oldid=27762
Retrieved from "https://encyclopediaofmath.org/index.php?title=User:Camillo.delellis/sandbox&oldid=30192"