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{{MSC|26A45}} (Functions of one variable)
  
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{{MSC|26B30|28A15,26B15,49Q15}} (Functions of severable variables)
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[[Category:Analysis]]
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{{TEX|done}}
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==Functions of one variable==
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===Classical definition===
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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'''
 +
Let  $I\subset \mathbb R$ be an interval and consider the collection $\Pi$  of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots <  a_N<b_N\in I$,
 +
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}
 +
TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\,
 +
\end{equation}
 +
(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
 +
\[
 +
TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, ,
 +
\]
 +
where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.
 +
 +
As a corollary we derive
 +
 +
'''Proposition 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.
 +
 +
===General properties===
 +
====Jordan decomposition====
 +
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
 +
 +
'''Theorem 3'''
 +
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.
 +
 +
(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.
 +
 +
'''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
 +
\[
 +
f (x^+) :=\lim_{y\downarrow x} f (y) \qquad f (x^-):= \lim_{y\uparrow x} f(y)
 +
\]
 +
exist at every point $x\in I$;
 +
* 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
 +
\[
 +
f (x) =\left\{\begin{array}{ll}
 +
1 \qquad &\mbox{if $x=0$}\\
 +
0 \qquad &\mbox{otherwise}
 +
\end{array}\right.
 +
\]
 +
is a function of bounded variation
 +
 +
====Precise representative====
 +
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 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====
 +
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
 +
\[
 +
f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt
 +
\]
 +
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'''
 +
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.
 +
 +
'''Theorem 9'''
 +
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
 +
* $\mu$ is the derivative, in the sense of distributions, of $f$, i.e. \eqref{e:distrib} holds
 +
* $F_\mu = \tilde{f} = f$ almost everywhere
 +
* $\tilde{f}$ is a function of bounded variation in the sense of '''Definition 1'''
 +
*  $TV (\tilde{f})$ equals the total variation of the measure $\mu$ which  in turn is equal to the supremum in \eqref{e:variation_modern}.
 +
 +
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.
 +
 +
===Structure 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.
 +
\[
 +
\mu_c (\{x\}) = 0\qquad \mbox{for every $x\in I$}\,
 +
\]
 +
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'''
 +
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.
 +
* If $J$ denotes the set of points of discontinuity of $f$, then
 +
\[
 +
\mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, .
 +
\]
 +
* At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f(x) = g(x)$.
 +
 +
====Lebesgue decomposition====
 +
Observe also that, if we define the functions
 +
* $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
 +
* $f_j (x) := \mu_j (]a, x])$,
 +
* $f_c (x) := \mu_c (]a, x])$,
 +
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}
 +
 +
====Absolutely continuous functions====
 +
[[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]]
 +
\[
 +
{\bf 1}_{[a, \infty[} (x) :=
 +
\left\{\begin{array}{ll}
 +
0 \qquad &\mbox{if $x<a$}\\
 +
1 \qquad &\mbox{if $x\geq a$}
 +
\end{array}\right.
 +
\]
 +
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.
 +
 +
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
 +
\[
 +
f = \sum_i c_i {\bf 1}_{[a_i, \infty[}\, .
 +
\]
 +
 +
====Cantor ternary function====
 +
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]].
 +
 +
===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}
 +
 +
As a consequence of the [[Radon-Nikodym theorem]] we then have
 +
 +
'''Prposition 14'''
 +
A  function $u\in L^1 (\Omega)$ is a function of bounded variation if and  only if $V(u, \Omega)<\infty$ and moreover $V (u,\Omega) = |Du|  (\Omega)$.
 +
 +
====Consistency with the one variable theory====
 +
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====
 +
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
 +
by Ambrosio in {{Cite|Am}}:
 +
 +
'''Definition 14'''
 +
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''').
 +
 +
===Functional properties===
 +
The space $BV (\Omega)$ enjoys several properties that are typical of the [[Sobolev space|Sobolev spaces]] $W^{1,p} (\Omega)$.
 +
====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====
 +
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}}.
 +
However,  differently from the usual Sobolev spaces, the space $C^\infty  (\Omega)$ is ''not dense'' in the strong topology: its closure is  instead $W^{1,1} (\Omega)$.
 +
====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}}).
 +
====Extension theorems====
 +
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'''
 +
Assume $\Omega$ is open and bounded, with $\partial \Omega$ of class $C^1$. Then there exists a bounded linear mapping
 +
\[
 +
T:BV (\Omega)\to L^1 (\partial \Omega, \mathcal{H}^{n-1})
 +
\]
 +
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 $W^{1,1} (\Omega)$.
 +
 +
===Pointwise properties===
 +
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}}
 +
====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
 +
\[
 +
u^+ (x) = {\rm ap} \lim_{y\in B^+, y \to x} u(y)
 +
\]
 +
\[
 +
u^- (x) = {\rm ap} \lim_{y\in B^-, y \to x} u(y)
 +
\]
 +
(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'''
 +
The precise representative of $u\in BV (\Omega)$ is the Borel measurable function defined by
 +
\[
 +
\tilde{u} (x) =\left\{
 +
\begin{array}{ll}
 +
{\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if $x\not\in S_u$}\\
 +
\frac{u^+ (x) + u^- (x)}{2} &\mbox{if $x\in J_u$,}
 +
\end{array}\right.
 +
\]
 +
which coincides with $u$ $\lambda$-a.e..
 +
====Rectifiability of the jump set====
 +
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'''
 +
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==
 +
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
 +
\[
 +
\Omega_x:=\{t\in\mathbb R: x+t\nu\in\Omega\}
 +
\]
 +
and we define the sections $u_x:\Omega_x\to\mathbb R$ as $u_x (t):= u (x+t\nu)$. We then have
 +
 +
'''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
 +
 +
'''Definition 21'''
 +
Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as
 +
\[
 +
V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\,
 +
\]
 +
and the Tonelli-Cesari variation as
 +
\[
 +
V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\mbox{$\lambda$-a.e.}\right\}\, .
 +
\]
 +
 +
'''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
 +
\[
 +
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 +
\]
 +
 +
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
 +
\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}
 +
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
 +
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 $D{\bf 1}_E =\mu$.
 +
 +
A possible (and quite common) alternative definition of perimeter is
 +
\[
 +
\inf  \left\{ \liminf_n\; \mathcal{H}^{n-1} (\partial E_k):\;\mbox{$\{E_k\}$  is a sequence of smooth sets with $\lambda (E\bigtriangleup E_k) \to  0$}\right\}\, .
 +
\]
 +
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===
 +
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
 +
\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'''
 +
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}}.
 +
===Reduced boundary and structure theorem===
 +
The essential boundary of a Caccioppoli set can be analyzed further.
 +
 +
'''Definition 25'''
 +
If $E\subset\Omega$ is a Caccioppoli set the ''reduced boundary'' of $E$ is defined as
 +
\[
 +
\mathcal{F}  E := \left\{ x\in\Omega : \nu_E (x) := \lim_{r\downarrow 0}  \frac{D{\bf  1}_E (B_r(x))}{|D {\bf 1}_E| (B_r(x))}\;\; \mbox{exists and  $|\nu_E  (x)|=1$}\right\}\, .
 +
\]
 +
$\nu_E$ is called the ''measure theoretic'' inner normal.
 +
 +
We then have the following fundamental result, due to De Giorgi (for a proof see Section 3.5 of {{Cite|AFP}}).
 +
 +
'''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'''
 +
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==
 +
{|
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|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.
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|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.
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|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}}
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|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
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|valign="top"|{{Ref|Co}}||  C. M. Dafermos, "Hyperbolic conservation laws in continuum physics",  2nd edition, Springer Verlag, 2005,
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|valign="top"|{{Ref|DM}}|| G. Dal Maso, "An introduction to $\Gamma$-convergence", Burkhäuser, 1993.
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|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.
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|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.
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|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.
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|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
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|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}}
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|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}}
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|valign="top"|{{Ref|Fi}}||  G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico  dell'Università di Trieste, vol. I, 1954.
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|valign="top"|{{Ref|FR}}||  W. H. Fleming, R. Rishel, "An integral formula for total gradient  variation", Arch. Math., '''11''' (1960) pp. 218-222.
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|-
 +
|}

Revision as of 17:15, 27 August 2012

2010 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)

2010 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)

Functions of one variable

Classical definition

Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if its total variation is bounded. The total variation is defined in the following way.

Definition 1 Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$, 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} TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\, \end{equation} (cp. with Section 4.4 of [Co] or Section 10.2 of [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 $(X,d)$: 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 map, then $\varphi\circ f$ is also a function of bounded variation and \[ TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, , \] where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.

As a corollary we derive

Proposition 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.

General properties

Jordan decomposition

A fundamental characterization of functions of bounded variation of one variable is due to Jordan.

Theorem 3 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.

(Cp. with Theorem 4 of Section 5.2 in [Ro]). Indeed it is possible to find a canonical representation of any function of bounded variation as difference of nondecreasing functions.

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 [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

\[ f (x^+) :=\lim_{y\downarrow x} f (y) \qquad f (x^-):= \lim_{y\uparrow x} f(y) \] exist at every point $x\in I$;

  • 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 \[ f (x) =\left\{\begin{array}{ll} 1 \qquad &\mbox{if '"`UNIQ-MathJax34-QINU`"'}\\ 0 \qquad &\mbox{otherwise} \end{array}\right. \] is a function of bounded variation

Precise representative

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 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 [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

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 [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 \[ f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt \] 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 measures. More precisely, consider a signed measure $\mu$ on (the 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 [Co]. Obvious generalizations hold in the case of different domains of definition.

Distributional derivatives: modern definition

The measure $\mu$ is indeed the 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 [EG].

Definition 8 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 [AFP] for a proof.

Theorem 9 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

  • $\mu$ is the derivative, in the sense of distributions, of $f$, i.e. \eqref{e:distrib} holds
  • $F_\mu = \tilde{f} = f$ almost everywhere
  • $\tilde{f}$ is a function of bounded variation in the sense of Definition 1
  • $TV (\tilde{f})$ equals the total variation of the measure $\mu$ which in turn is equal to the supremum in \eqref{e:variation_modern}.

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.

Structure 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 [AFP] and decompose $\mu_s = \mu_c +\mu_j$, where $\mu_c$ is the non-atomic part of the measure $\mu_s$, i.e. \[ \mu_c (\{x\}) = 0\qquad \mbox{for every '"`UNIQ-MathJax76-QINU`"'}\, \] 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 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 [AFP]), which is often referred to as BV structure theorem fur functions of one variable.

Theorem 10 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.

  • If $J$ denotes the set of points of discontinuity of $f$, then

\[ \mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, . \]

  • At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f(x) = g(x)$.

Lebesgue decomposition

Observe also that, if we define the functions

  • $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
  • $f_j (x) := \mu_j (]a, x])$,
  • $f_c (x) := \mu_c (]a, x])$,

then

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}

Absolutely continuous functions

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 absolutely continuous measure.

Jump functions

The indicator function of the half line, also called Heaviside function \[ {\bf 1}_{[a, \infty[} (x) := \left\{\begin{array}{ll} 0 \qquad &\mbox{if '"`UNIQ-MathJax102-QINU`"'}\\ 1 \qquad &\mbox{if '"`UNIQ-MathJax103-QINU`"'} \end{array}\right. \] 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.

The Heaviside function is a prototype of 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 \[ f = \sum_i c_i {\bf 1}_{[a_i, \infty[}\, . \]

Cantor ternary function

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 [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 in the Lebesgue decomposition.

Historical remark

Functions of bounded variation were introduced for the first time by C. Jordan in [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 [Ed]. The criterion is also called Jordan-Dirichlet test, see [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 [DG] and [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 [To], proposed by Cesari [Ce], cp. with the section Tonelli-Cesari variation below. We refer to Section 3.12 of [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 currents in $\mathbb R^n$. This is the point of view of Federer, [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 [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 partial derivatives of $u$ in the sense of distributions are 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 $\sigma$-algebra of 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 [AFP]).

Total variation

Some authors use instead the following alternative road (cp. with Section 5.1 of [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}

As a consequence of the Radon-Nikodym theorem we then have

Prposition 14 A function $u\in L^1 (\Omega)$ is a function of bounded variation if and only if $V(u, \Omega)<\infty$ and moreover $V (u,\Omega) = |Du| (\Omega)$.

Consistency with the one variable theory

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

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 by Ambrosio in [Am]:

Definition 14 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).

Functional properties

The space $BV (\Omega)$ enjoys several properties that are typical of the Sobolev spaces $W^{1,p} (\Omega)$.

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 [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 [EG]).

Semicontinuity of the variation

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 [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 [AFP]. However, differently from the usual Sobolev spaces, the space $C^\infty (\Omega)$ is not dense in the strong topology: its closure is instead $W^{1,1} (\Omega)$.

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 [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 [AFP]).

Extension theorems

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 [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 [EG] or Theorem 3.47 of Section 3.4 of [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 $n\geq 2$, \[ \left(\int_{B_r (x)} |u (y)-\bar{u}|^{\frac{n-1}{n}}\right)^{\frac{n-1}{n}}\, \;\leq\; C (n) \, V (u, B_r (x)) \qquad \mbox{for every '"`UNIQ-MathJax245-QINU`"'} \] 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 [EG] or Remark 3.50 of Section 3.4 on [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 $n-1$-dimensional measure.

Theorem 16 Assume $\Omega$ is open and bounded, with $\partial \Omega$ of class $C^1$. Then there exists a bounded linear mapping \[ T:BV (\Omega)\to L^1 (\partial \Omega, \mathcal{H}^{n-1}) \] 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 [EG]). By a Theorem of Gagliardo, see [Ga], the trace operator is in fact onto, even when restricted to $W^{1,1} (\Omega)$.

Pointwise properties

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 [AFP] or in Section 5.9 of [EG]

Approximate continuity

There is a Borel set $S_u$ with $\sigma$-finite $\mathcal{H}^{n-1}$ measure such that $u$ the 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 \[ u^+ (x) = {\rm ap} \lim_{y\in B^+, y \to x} u(y) \] \[ u^- (x) = {\rm ap} \lim_{y\in B^-, y \to x} u(y) \] (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 The precise representative of $u\in BV (\Omega)$ is the Borel measurable function defined by \[ \tilde{u} (x) =\left\{ \begin{array}{ll} {\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if '"`UNIQ-MathJax286-QINU`"'}\\ \frac{u^+ (x) + u^- (x)}{2} &\mbox{if '"`UNIQ-MathJax287-QINU`"',} \end{array}\right. \] which coincides with $u$ $\lambda$-a.e..

Rectifiability of the jump set

The set $J_u$ is 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 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 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 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 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

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 \[ \Omega_x:=\{t\in\mathbb R: x+t\nu\in\Omega\} \] and we define the sections $u_x:\Omega_x\to\mathbb R$ as $u_x (t):= u (x+t\nu)$. We then have

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 [EG] or Section 3.11 in [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 [Ce] as a modification of Tonelli's plabe variation. More precisely

Definition 21 Given a measurable function $f: \mathbb R^2\to\mathbb R$ we define the Tonelli variation $f$ as \[ V_T (f) := \int_{-\infty}^\infty TV (f (\cdot, y))\, dy + \int_{-\infty}^\infty TV (f (x, \cdot))\, dx\, \] and the Tonelli-Cesari variation as \[ V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\mbox{$\lambda$-a.e.}\right\}\, . \] '''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 $V (f,\mathbb R^2)\leq V_{TC} (f) \leq \sqrt{2} V (f,\mathbb R^2)$. =='"`UNIQ--h-43--QINU`"'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 \[ {\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, . \] 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 \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} 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 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 $D{\bf 1}_E =\mu$. A possible (and quite common) alternative definition of perimeter is \[ \inf \left\{ \liminf_n\; \mathcal{H}^{n-1} (\partial E_k):\;\mbox{$\{E_k\}$ is a sequence of smooth sets with $\lambda (E\bigtriangleup E_k) \to 0$}\right\}\, . \] 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 [[#Ca|[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. ==='"`UNIQ--h-44--QINU`"'Characterization through density=== 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 \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''' 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 [[#AFP|[AFP]]]) and by $\partial_* E$ by others (see [[#EG|[EG]]]). See Theorem 3.61 in [[#AFP|[AFP]]]. In what follows we will stcik to the notation of [[#AFP|[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 [[#EG|[EG]]]. ==='"`UNIQ--h-45--QINU`"'Reduced boundary and structure theorem=== The essential boundary of a Caccioppoli set can be analyzed further. '''Definition 25''' If $E\subset\Omega$ is a Caccioppoli set the ''reduced boundary'' of $E$ is defined as \[ \mathcal{F} E := \left\{ x\in\Omega : \nu_E (x) := \lim_{r\downarrow 0} \frac{D{\bf 1}_E (B_r(x))}{|D {\bf 1}_E| (B_r(x))}\;\; \mbox{exists and $|\nu_E (x)|=1$}\right\}\, . \] $\nu_E$ is called the ''measure theoretic'' inner normal. We then have the following fundamental result, due to De Giorgi (for a proof see Section 3.5 of [[#AFP|[AFP]]]). '''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} ==='"`UNIQ--h-46--QINU`"'Generalized divergence theorem=== '''Theorem 26''' can also be interpreted as a far-reaching generalization of the divergence theorem. We have namely '''Corollary 27''' 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! ==='"`UNIQ--h-47--QINU`"'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 [[#DG2|[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 [[#AFP|[AFP]]]. ==='"`UNIQ--h-48--QINU`"'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 [[#FR|[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 [[#AFP|[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. =='"`UNIQ--h-49--QINU`"'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 [[#AFP|[AFP]]]). A theorem due to Volpert (see [[#Vo|[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 [[#AFP|[AFP]]]). The chain rule of Volpert has been generalized by Ambrosio and Dal Maso to Lipschitz $\varphi$ (see [[#AD|[AD]]]). =='"`UNIQ--h-50--QINU`"'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 ([[#Al|[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 [[#DL|[DL]]]. =='"`UNIQ--h-51--QINU`"'Special Functions of bounded variation== In [[#DA|[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 [[#AFP|[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 [[#AFP|[AFP]]] for a comprehensive account of the theory of special functions of bounded variation. =='"`UNIQ--h-52--QINU`"'Notable applications== ==='"`UNIQ--h-53--QINU`"'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 [[#Gi|[Gi]]] for a quite thorough account. ===='"`UNIQ--h-54--QINU`"'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. ==='"`UNIQ--h-55--QINU`"'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 [[#Br|[Br]]], [[#Da|[Da]]] and [[#Se|[Se]]]. ==='"`UNIQ--h-56--QINU`"'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. ==='"`UNIQ--h-57--QINU`"'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 [[#DM|[DM]]]. =='"`UNIQ--h-58--QINU`"'References== {| |- |valign="top"|<span id="Al"></span>[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"|<span id="Am"></span>[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"|<span id="AD"></span>[AD]|| L. Ambrosio, G. Dal Maso, "A general chain rule for distributional derivatives", Proc. Amer. Math. Soc., '''108''' (1990) pp. 691-792. |- |valign="top"|<span id="AFP"></span>[AFP]|| L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. 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How to Cite This Entry:
Function of bounded variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_bounded_variation&oldid=27764
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article