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{{MSC|26A45}} (Functions of one variable)
 
{{MSC|26A45}} (Functions of one variable)
  
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===Classical definition===
 
===Classical definition===
 
Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if
 
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.
+
its {{Anchor|Total variation}} [[Variation of a function|total variation]] is bounded. The total variation is defined in the following way.
  
 
'''Definition 1'''
 
'''Definition 1'''
Let  $I\subset \mathbb R$ be an interval and consider the collection $\Pi$  of ordered $2N$-ples of points $a_1<b_1<a_2< b_2 < \ldots <  a_N<b_N\in I$,
+
Let  $I\subset \mathbb R$ be an interval and consider the collection $\Pi$  of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots <  a_{N+1}\in I$,
 
where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by
 
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}
 
\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\}\,  
+
TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(a_{i+1})-f(a_i)| : (a_1, \ldots, a_{N+1})\in\Pi\right\}\,  
 
\end{equation}
 
\end{equation}
 
(cp. with Section 4.4 of {{cite|Co}} or Section 10.2 of {{Cite|Ro}}).
 
(cp. with Section 4.4 of {{cite|Co}} or Section 10.2 of {{Cite|Ro}}).
  
 
====Generalizations====
 
====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
+
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 $|f(a_{i+1})-f(a_i)|$ with $d (f(a_{i+1}),  f(a_i))$ 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)\, ,
 
TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, ,
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===General properties===
 
===General properties===
 
====Jordan decomposition====
 
====Jordan decomposition====
 +
{{Anchor|Jordan decomposition}}
 
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
 
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
  
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\[
 
\[
 
f (x) =\left\{\begin{array}{ll}
 
f (x) =\left\{\begin{array}{ll}
1 \qquad &\mbox{if $x=0$}\\
+
1 \qquad &\mbox{if } x=0\\
 
0 \qquad &\mbox{otherwise}
 
0 \qquad &\mbox{otherwise}
 
\end{array}\right.
 
\end{array}\right.
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====Precise representative====
 
====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
+
In  order to avoid pathologies 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}};
 
* 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 $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.
 
* at any point $x$ we impose $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.
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====Distributional derivatives: modern 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
+
The  measure $\mu$ is indeed the [[Generalized derivative|generalized  derivative]] of the function $f=F_\mu$ in the sense of distributions.  More precisely
 
\begin{equation}\label{e:distrib}
 
\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)\, .
 
\int f(t)\varphi' (t)\, dt = -\int \varphi (t)\, d\mu (t) \qquad \forall \varphi\in C^\infty_c (\mathbb R)\, .
 
\end{equation}
 
\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}}.
+
This identity 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'''
 
'''Definition 8'''
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===Structure theorem===
 
===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.
+
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 $\lambda$. 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 $x\in I$}\,  
+
\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
 
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
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\mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, .
 
\mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, .
 
\]
 
\]
* At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f(x) = g(x)$.  
+
* At $\lambda$-a.e. $x$ the function $f$ is differentiable and $f'(x) = g(x)$.  
  
 
====Lebesgue decomposition====
 
====Lebesgue decomposition====
Observe also that, if we define the functions
+
{{Anchor|Lebesgue decomposition}} Observe also that, if we define the functions
 
* $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
 
* $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
 
* $f_j (x) := \mu_j (]a, x])$,
 
* $f_j (x) := \mu_j (]a, x])$,
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* $f_c$ is a [[Singular function|singular function]]  
 
* $f_c$ is a [[Singular function|singular function]]  
 
* $f_j$ is a [[Jump function|jump 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
+
Then  $f=f_a+f_c+f_j$ is called the [[Lebesgue decomposition]] of the  function $f$ and it is unique up to constants.
  
 
===Examples===
 
===Examples===
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{\bf 1}_{[a, \infty[} (x) :=  
 
{\bf 1}_{[a, \infty[} (x) :=  
 
\left\{\begin{array}{ll}
 
\left\{\begin{array}{ll}
0 \qquad &\mbox{if $x<a$}\\
+
0 \qquad &\mbox{if } x<a\\
1 \qquad &\mbox{if $x\geq a$}
+
1 \qquad &\mbox{if } x\geq a
 
\end{array}\right.
 
\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.
+
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 variation is $1$, 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
 
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
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====Cantor ternary function====
 
====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]].
+
The  [[Cantor ternary function]], also called Devil's staircase (and  Cantor-Vitali function, 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===
 
===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]].
+
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 the [[Dirichlet  theorem]] on the convergence of Fourier series, called [[Jordan  criterion]].
  
 
'''Theorem 11'''
 
'''Theorem 11'''
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==Functions of several variables==
 
==Functions of several variables==
 
===Historical remarks===
 
===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.  
+
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  approaches 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, Federer and Volpert. 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====
 
====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,
+
Functions  of bounded variation in $\mathbb R^n$ can be identified with  $n$-dimensional normal [[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===
 
===Definition===
 
Following Section 3.1 of {{Cite|AFP}},
 
Following Section 3.1 of {{Cite|AFP}},
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The vector space of all functions of bounded variations on $\Omega$ is denoted by $BV (\Omega)$.  
 
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}}).  
+
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 space $BV_{loc} (\Omega)$ 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====
 
====Total variation====
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====Consistency with the one variable theory====
 
====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.
+
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 $V(f, I)<\infty$, 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====
 
====Generalizations====
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*  There is a measure $\mu$ such that $|D (\varphi\circ f)|\leq {\rm  Lip}\, (\varphi) \mu$ for every Lipschitz function $\varphi:X\to\mathbb  R$.
 
*  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''').  
+
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===
 
===Functional properties===
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====Semicontinuity of the variation====
 
====Semicontinuity of the variation====
If a sequence of functions $\{u_n\}\in L^1 (\Omega)$ converges strongly to $L^1 (\Omega)$, then  
+
If a sequence of functions $\{u_n\}\subset L^1 (\Omega)$ converges strongly to $u\in L^1 (\Omega)$, then  
 
\[
 
\[
 
\liminf_{n\to\infty}\, V (u_n, \Gamma)\geq V (u, \Gamma)
 
\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 $u\in BV (\Omega)$.
 
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====
 
====Approximation with smooth functions====
 
'''Theorem 15'''
 
'''Theorem 15'''
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* $\|u_n-u\|_{L^1 (\Omega)} \to 0$
 
* $\|u_n-u\|_{L^1 (\Omega)} \to 0$
 
* $\liminf_n V (u_n, \Omega) < \infty$.
 
* $\liminf_n V (u_n, \Omega) < \infty$.
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 $\|u_n\|_{BV}\to  \|u\|_{BV}$.
+
Moreover,  for every $u\in BV (\Omega)$ there is an approximating sequence  $\{u_n\}\subset C^\infty\cap BV (\Omega)$ converging strongly to $u$ in $L^1$ 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}}.
 
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 $W^{1,1} (\Omega)$.
+
However,  differently from the usual Sobolev spaces, the space $C^\infty  (\Omega)$ is ''not dense'' in the strong topology: its strong closure is  instead $W^{1,1} (\Omega)$.
 +
 
 
====Weak$^\star$ convergence====
 
====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}}
+
A  sequence $\{u_n\}$ converges weakly$^\star$ in $BV (\Omega)$ to $u$ if  $u_h\to u$ 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}}
 +
 
 +
Moreover, closed and bounded convex subsets of $BV (\Omega)$ are  weakly$^\star$ compact if $\Omega$ is bounded (cp. with Theorem 3.23 in Section 3.1 of  {{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====
 
====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.
 
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====
 
====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:
+
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 $f\in BV (\mathbb R)$;
 
*$\|f\|_{L^\infty}\leq C(1) TV (f)$ 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 $n\geq 2$.
 
*$\|f\|_{L^{n/(n-1)}}\leq V (u,\mathbb R^n)$ 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.
+
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|AFP}}.
 
====Poincaré inequality====
 
====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 $n\geq 2$,   
 
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 $u\in BV (B_r (x))$}
+
\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  
 
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
 
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 $\Omega$.
 
specific geometry of $\Omega$.
 +
 
====Trace operator====
 
====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]].
 
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]].
Line 289: Line 296:
  
 
===Pointwise properties===
 
===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}}
+
In  the following sections 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====
 
====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'' $x\not\in S_u$.  
+
There  is a Borel set $S_u$ with $\sigma$-finite [[Hausdorff  measure|$\mathcal{H}^{n-1}$ measure]] such that the [[Approximate  limit|approximate limit]] of $u$ exists at ''every'' $x\not\in S_u$.
 +
 
 
====Jump set====
 
====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
+
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\}\, ,
+
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
 
then
Line 305: Line 313:
 
\]
 
\]
 
(for the definition of ${\rm ap}\lim$ see [[Approximate limit]]).
 
(for the definition of ${\rm ap}\lim$ see [[Approximate limit]]).
 +
 
====Precise representative====
 
====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,  
 
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,  
Line 313: Line 322:
 
\tilde{u} (x) =\left\{
 
\tilde{u} (x) =\left\{
 
\begin{array}{ll}
 
\begin{array}{ll}
{\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if $x\not\in S_u$}\\
+
{\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if } x\not\in S_u\\
\frac{u^+ (x) + u^- (x)}{2} &\mbox{if $x\in J_u$,}
+
\frac{u^+ (x) + u^- (x)}{2} &\mbox{if } x\in J_u\, ,
 
\end{array}\right.
 
\end{array}\right.
 
\]
 
\]
which coincides with $u$ $\lambda$-a.e..  
+
which coincides with $u$ $\lambda$-a.e..
 +
 
 
====Rectifiability of the jump set====
 
====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  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).
Line 324: Line 334:
 
$u$  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]].  
 
$u$  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====
 
====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
+
{{Anchor|Structure theorem several}} 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'''
 
'''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
+
According  to the [[Radon-Nikodym decomposition]] $Du$ 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 $\mathcal{H}^{n-1} (E) <\infty$;
 
* $Du^c (E) =0$ for every Borel set with $\mathcal{H}^{n-1} (E) <\infty$;
 
* For any Borel set $E$ we have the identity
 
* For any Borel set $E$ we have the identity
Line 333: Line 343:
 
Du^j (E) = \int_{E\cap J_u} (u^+ (x)-u^-(x))\, \nu (x)\, d\mathcal{H}^{n-1} (x)\, .
 
Du^j (E) = \int_{E\cap J_u} (u^+ (x)-u^-(x))\, \nu (x)\, d\mathcal{H}^{n-1} (x)\, .
 
\end{equation}
 
\end{equation}
 +
 
====Vector-valued case====
 
====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  
 
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  
Line 357: Line 368:
  
 
'''Theorem 20'''
 
'''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'''.
+
Let  $u\in BV (\Omega)$ and let $\tilde{u}$ be 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
+
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  plane variation]]. More precisely
  
 
'''Definition 21'''
 
'''Definition 21'''
Line 368: Line 379:
 
and the Tonelli-Cesari variation as
 
and the Tonelli-Cesari variation as
 
\[
 
\[
V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\mbox{$\lambda$-a.e.}\right\}\, .
+
V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\lambda\mbox{-a.e.}\right\}\, .
 
\]
 
\]
  
Line 380: Line 391:
  
 
'''Definition 23'''
 
'''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
+
Let  $\Omega\subset \mathbb R^n$ be an open set and $E\subset \Omega$ a  measurable set such that $ V({\bf 1}_E, \Omega)<\infty$. The $E$ is called a  ''Caccioppoli set'' or a ''set of finite perimeter'' and its perimeter  in $\Omega$ is defined to be
 
\[
 
\[
 
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 
\]
 
\]
  
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
+
'''Warning''' Since it is sometimes convenient to consider unbounded Caccioppoli sets, we will not assume that the set $E$ has finite measure: note that anyway the quantity $V({\bf 1}_E, \Omega)$ is well defined.
 +
 
 +
A  primary example is given by those open sets $E\subset \Omega$ which  have a $C^1$ topological boundary $\partial E$ with $\mathcal{H}^{n-1} ((\partial E)\cap\Omega) < \infty$. If we denote by $\nu$ the exterior unit normal field at $\partial E$, the [[Divergence|divergence theorem]]  
 +
implies
 
\begin{equation}\label{e:divergenza1}
 
\begin{equation}\label{e:divergenza1}
 
\int  {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\,  \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\,  d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\,  .
 
\int  {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\,  \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\,  d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\,  .
 
\end{equation}
 
\end{equation}
It turns then out by that $V ({\bf 1}_E,  \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$, see '''Definition  13''', and hence ${\bf 1}_E\in BV_{loc} (\Omega)$ (for having ${\bf  1}_E\in BV (\Omega)$ we need the additional condition $\lambda  (E)<\infty$). In particular, if
+
Thus $V ({\bf 1}_E,  \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$, 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
 
we introduce the vector measure
 
\[
 
\[
Line 398: Line 412:
 
A possible (and quite common) alternative definition of perimeter is
 
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\}\, .
+
\inf  \left\{ \liminf_k\; \mathcal{H}^{n-1} (\partial E_k):\;\{E_k\} \mbox{ is a sequence of smooth sets with } \lambda (E\bigtriangleup E_k) \to  0\right\}\, .
 
\]
 
\]
 
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.
 
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.
Line 412: Line 426:
 
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}}).
 
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}}.
+
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, in the following sense: if $E\subset \Omega$ is a measurable set and $\mathcal{H}^{n-1} (K\setminus \{x: \theta^n (E,x)\in \{0,1\})< \infty$ for every compact subset $K\subset \Omega$, then $E$ is a Caccioppoli set. The latter is a deep  theorem by Federer: see Theorem 4.5.11 of {{Cite|Fe}} and Section 5.11 of {{Cite|EG}}.
 +
Observe on the other hand that the following open subset of the real line
 +
\[
 +
E := \bigcup_{k=1}^\infty ]2^{-k}, 2^{-k} + 3^{-k}[
 +
\]
 +
is not a Caccioppoli set, although the density $\theta^1 (E,x)$ is either $0$, $\frac{1}{2}$ or $1$ at every point $x\in \mathbb R$.
 +
 
 
===Reduced boundary and structure theorem===
 
===Reduced boundary and structure theorem===
 
The essential boundary of a Caccioppoli set can be analyzed further.
 
The essential boundary of a Caccioppoli set can be analyzed further.
Line 419: Line 439:
 
If $E\subset\Omega$ is a Caccioppoli set the ''reduced boundary'' of $E$ is defined as
 
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\}\, .
+
\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.
 
$\nu_E$ is called the ''measure theoretic'' inner normal.
Line 426: Line 446:
  
 
'''Theorem 26'''
 
'''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
+
For  any $x\in \mathcal{F} E$ the Lebesgue density $\theta^n (E,x)$ is equal  to $\frac{1}{2}$, hence the reduced boundary is a subset of  the  essential boundary $\partial^* E$ and moreover $\mathcal{H}^{n-1}  (\partial^*  E\setminus\mathcal{F} E) = 0$. 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}
 
\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)\, .
 
D {\bf 1}_E (A) = \int_{A\cap \mathcal{F} E} \nu_E (x)\, d\mathcal{H}^{n-1} (x)\, .
 
\end{equation}
 
\end{equation}
 +
 
===Generalized divergence theorem===
 
===Generalized divergence theorem===
 
'''Theorem 26''' can also be interpreted as a far-reaching generalization of the divergence theorem. We have namely
 
'''Theorem 26''' can also be interpreted as a far-reaching generalization of the divergence theorem. We have namely
Line 450: Line 471:
 
Then  
 
Then  
 
\[
 
\[
\lambda (E) \leq C \Big({\rm Per}\, (E, \mathbb R^n)\Big)^{\alpha}\, .
+
\lambda (E) \leq C(n)\, \Big({\rm Per}\, (E, \mathbb R^n)\Big)^{\alpha}\, .
\qquad \mbox{for any Caccioppoli set $E\subset\mathbb R^n$}.
+
\qquad \mbox{for any Caccioppoli set } E\subset\mathbb R^n
 
\]
 
\]
 +
and the equality holds if and only if $E$ is a ball.
  
 
A ''relative isoperimetric'' inequality holds also in extension domains $\Omega$, see Exercise 3.13 of {{Cite|AFP}}.
 
A ''relative isoperimetric'' inequality holds also in extension domains $\Omega$, see Exercise 3.13 of {{Cite|AFP}}.
 +
 
===Coarea formula===
 
===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}}.
 
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}}.
Line 471: Line 494:
  
 
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.
 
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==
 
==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
 
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
Line 499: Line 523:
  
 
==Special Functions of bounded variation==
 
==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''').
+
In  {{Cite|DA}}, in order to study variational problems involving free  discontinuities (most notably the [[Mumford-Shah functional]]) De Giorgi  and Ambrosio considered the 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.
+
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, Ambrosio 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'''
 
'''Theorem 32'''
Line 516: Line 540:
 
==Notable applications==
 
==Notable applications==
 
===Plateau's problem===
 
===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.  
+
Since  their introduction 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====
 
====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.
+
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===
 
===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}}.
 
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===
+
===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.
+
$SBV (\Omega)$ 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===
 
===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
 
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
Line 529: Line 555:
 
\]
 
\]
 
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}}.  
 
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==
 
==References==
 
{|
 
{|
 
|-
 
|-
|valign="top"|{{Ref|Al}}||  G. Alberti, "Rank-one properties for derivatives of functions of  bounded variation", Proc. Roy Soc. Edinburgh Sect. A, '''123''' (1993)  pp. 239-274  
+
|valign="top"|{{Ref|Al}}||  G. Alberti, "Rank-one properties for derivatives of functions of  bounded variation", Proc. Roy Soc. Edinburgh Sect. A, '''123''' (1993)  pp. 239-274  
 
|-
 
|-
|valign="top"|{{Ref|Am}}|| L. Ambrosio,  "Metric space valued functions with bounded variation", Ann. Scuola  Norm. Sup. Pisa Cl. Sci. (4), '''17''' (1990) pp. 291-322.
+
|valign="top"|{{Ref|Am}}|| L. Ambrosio,  "Metric space valued functions with bounded variation", Ann. Scuola  Norm. Sup. Pisa Cl. Sci. (4), '''17''' (1990) pp. 291-322.  
 
|-
 
|-
|valign="top"|{{Ref|AD}}||  L. Ambrosio, G. Dal Maso, "A general chain rule for distributional  derivatives", Proc. Amer. Math. Soc., '''108''' (1990) pp. 691-792.
+
|valign="top"|{{Ref|AD}}||  L. Ambrosio, G. Dal Maso, "A general chain rule for distributional  derivatives", Proc. Amer. Math. Soc., '''108''' (1990) pp. 691-792. {{MR|0969514}}  {{ZBL|0685.49027}}
 
|-
 
|-
 
|valign="top"|{{Ref|AFP}}||    L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded  variations  and  free  discontinuity  problems". Oxford Mathematical  Monographs. The    Clarendon Press,  Oxford University Press, New York,  2000.      {{MR|1857292}}{{ZBL|0957.49001}}  
 
|valign="top"|{{Ref|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|Ca}}||  R. Caccioppoli, "Misura e integrazione sugli insiemei dimensionalmente  orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11  and 137-146.
 
|valign="top"|{{Ref|Ca}}||  R. Caccioppoli, "Misura e integrazione sugli insiemei dimensionalmente  orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf 12} (1952) pp. 3-11  and 137-146.
 
|-
 
|-
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+
|valign="top"|{{Ref|Ce}}|| L. Cesari, "Sulle  funzioni a variazione limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci.  (2), '''5''' (1936) pp. 299-313.  
 
|-
 
|-
 
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
 
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
 
|-
 
|-
|valign="top"|{{Ref|Co}}||  C. M. Dafermos, "Hyperbolic conservation laws in continuum physics",  2nd edition, Springer Verlag, 2005,
+
|valign="top"|{{Ref|Da}}||  C. M. Dafermos, "Hyperbolic conservation laws in continuum physics",  2nd edition, Springer Verlag, 2005,
 
|-
 
|-
 
|valign="top"|{{Ref|DM}}|| G. Dal Maso, "An introduction to $\Gamma$-convergence", Burkhäuser, 1993.
 
|valign="top"|{{Ref|DM}}|| G. Dal Maso, "An introduction to $\Gamma$-convergence", Burkhäuser, 1993.
 
|-
 
|-
|valign="top"|{{Ref|DG}}||  E. De Giorgi, L. Ambrosio, "Un nuovo funzionale nel calcolo delle  variazioni", Att. Acc. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8) Mat.  Appl., '''82''' (1988) pp. 199-210.
+
|valign="top"|{{Ref|DG}}||  E. De Giorgi, L. Ambrosio, "Un nuovo funzionale nel calcolo delle  variazioni", Att. Acc. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8) Mat.  Appl., '''82''' (1988) pp. 199-210.  
 
|-
 
|-
|valign="top"|{{Ref|DG}}||  E. De Giorgi, "Su una teoria generale della misura $n-1$-dimensionale  in uno spazio a $r$ dimensioni", Ann. Mat. Pura Appl. (4), '''36'''  (1954) pp. 191-213.
+
|valign="top"|{{Ref|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.   {{ZBL|0055.28504}}
 
|-
 
|-
|valign="top"|{{Ref|DG2}}|| E. De  Giorgi, "Sulla proprietà isoperimetrica dell'ipersfera, nella classe  degli insiemi aventi frontiera orientata di misura finita", Att. Acc.  Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, '''8''' (1958) pp.  33-44.
+
|valign="top"|{{Ref|DG2}}|| E. De  Giorgi, "Sulla proprietà isoperimetrica dell'ipersfera, nella classe  degli insiemi aventi frontiera orientata di misura finita", Att. Acc.  Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, '''8''' (1958) pp.  33-44.  
 
|-
 
|-
 
|valign="top"|{{Ref|DL}}|| C. De Lellis, "A note  on Alberti's rank-one theorem", Transport equations and multi-D  hyperbolic conservation laws, 61-74, Lect. Notes Unione Mat. Ital., 5,  Springer, Berlin, 2008.
 
|valign="top"|{{Ref|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|Fi}}||  G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico  dell'Università di Trieste, vol. I, 1954.
 
|valign="top"|{{Ref|Fi}}||  G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico  dell'Università di Trieste, vol. I, 1954.
 
|-
 
|-
|valign="top"|{{Ref|FR}}||  W. H. Fleming, R. Rishel, "An integral formula for total gradient  variation", Arch. Math., '''11''' (1960) pp. 218-222.
+
|valign="top"|{{Ref|FR}}||  W. H. Fleming, R. Rishel, "An integral formula for total gradient  variation", Arch. Math., '''11''' (1960) pp. 218-222. {{MR|0114892}}  {{ZBL|0094.26301}}
 
|-
 
|-
|valign="top"|{{Ref|Ga}}||  E. Gagliardo, "Caratterizzazione delle tracce sulla frontiera relative  ad alcune classi di funzioni in piú variabili", Rend. Sem. Mat. Univ.  Padova, '''27''' (1957) pp. 284-305.  
+
|valign="top"|{{Ref|Ga}}||  E. Gagliardo, "Caratterizzazione delle tracce sulla frontiera relative  ad alcune classi di funzioni in piú variabili", Rend. Sem. Mat. Univ.  Padova, '''27''' (1957) pp. 284-305.  
 
|-
 
|-
 
|valign="top"|{{Ref|Gi}}|| E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
 
|valign="top"|{{Ref|Gi}}|| E. Giusti, "Minimal surfaces and functions of bounded variation", Birkhäuser, 1994.
 
|-
 
|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
+
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}  
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|valign="top"|{{Ref|HS}}||    E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" ,  Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}} 
 
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|valign="top"|{{Ref|HS}}||   E. HewittK.R. Stromberg,  "Real and abstract analysis" Springer  (1965) {{MR|0188387}} {{ZBL|0137.03202}}
+
|valign="top"|{{Ref|Jo}}|| C. Jordan"Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92''' (1881) pp. 228–230  JFM {{ZBL|13.0184.01}}  
 
|-
 
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|valign="top"|{{Ref|Jo}}|| C. Jordan,  "Sur la série de Fourier" ''C.R. Acad. Sci. Paris'' , '''92''' (1881pp. 228–230
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|valign="top"|{{Ref|Le}}|| H. Lebesgue,  "Leçons sur l'intégration  et la récherche des fonctions primitives", Gauthier-Villars (1928). {{MR|2857993}}  JFM {{ZBL|54.0257.01}}
 
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}  
 
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|valign="top"|{{Ref|Se}}|| D. Serre, "Systems of conservation laws I/II". Cambridge University Press, 1999.  
 
|valign="top"|{{Ref|Se}}|| D. Serre, "Systems of conservation laws I/II". Cambridge University Press, 1999.  
 
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|valign="top"|{{Ref|To}}||  L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione  limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp.  315-320.
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|valign="top"|{{Ref|To}}||  L. Tonelli, "Sulle funzioni di due variabili generalmente a variazione  limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), '''5''' (1936) pp.  315-320.  
 
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|}

Latest revision as of 20:25, 12 March 2020

2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)

2020 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 $(N+1)$-ples of points $a_1<a_2 < \ldots < a_{N+1}\in I$, 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(a_{i+1})-f(a_i)| : (a_1, \ldots, a_{N+1})\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 $|f(a_{i+1})-f(a_i)|$ with $d (f(a_{i+1}), f(a_i))$ 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 given 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 pathologies 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 $\mu$ (i.e. $|\mu| (\mathbb R))$.
  • For every right-continuous function $f:\mathbb R\to \mathbb R$ of bounded variation with $\lim_{x\to-\infty} f (x) = 0$ 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 derivative 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 identity is the starting point for the modern definition of functions of bounded variation, cp. with [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 $\lambda$. 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. \[ \mu_c (\{x\}) = 0\qquad \mbox{for every } x\in I\, \] 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 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.

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 \[ {\bf 1}_{[a, \infty[} (x) := \left\{\begin{array}{ll} 0 \qquad &\mbox{if } x<a\\ 1 \qquad &\mbox{if } x\geq a \end{array}\right. \] is a function of bounded variation (on $\mathbb R$) with total variation equal to $1$. Its generalized derivative is the Dirac mass $\delta_a$. Obviously the Heaviside function is differentiable a.e. with derivative $0$ but its total variation is $1$, 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 function, 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 the 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 approaches 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, Federer and Volpert. 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 bounded variation in $\mathbb R^n$ can be identified with $n$-dimensional normal 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 space $BV_{loc} (\Omega)$ 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 $V(f, I)<\infty$, 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 $\{u_n\}\subset L^1 (\Omega)$ converges strongly to $u\in L^1 (\Omega)$, 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 $\{u_n\}\subset C^\infty\cap BV (\Omega)$ converging strongly to $u$ in $L^1$ 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 strong closure is instead $W^{1,1} (\Omega)$.

Weak$^\star$ convergence

A sequence $\{u_n\}$ converges weakly$^\star$ in $BV (\Omega)$ to $u$ if $u_h\to u$ 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]

Moreover, closed and bounded convex subsets of $BV (\Omega)$ are weakly$^\star$ compact if $\Omega$ is bounded (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 [AFP].

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 [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 the following sections 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 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 \[ \tilde{u} (x) =\left\{ \begin{array}{ll} {\rm ap}\lim_{y\to x} u (y)\qquad &\mbox{if } x\not\in S_u\\ \frac{u^+ (x) + u^- (x)}{2} &\mbox{if } x\in J_u\, , \end{array}\right. \] which coincides with $u$ $\lambda$-a.e..

Rectifiability of the jump set

The set $J_u$ is rectifiable, 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 decomposition $Du$ 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 let $\tilde{u}$ be 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 [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 plane 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 \[ V_{TC} (f) := \inf \left\{ V_T (g) : g = f \;\lambda\mbox{-a.e.}\right\}\, . \]

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 $ V({\bf 1}_E, \Omega)<\infty$. The $E$ is called a Caccioppoli set or a set of finite perimeter and its perimeter in $\Omega$ is defined to be \[ {\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, . \]

Warning Since it is sometimes convenient to consider unbounded Caccioppoli sets, we will not assume that the set $E$ has finite measure: note that anyway the quantity $V({\bf 1}_E, \Omega)$ is well defined.

A primary example is given by those open sets $E\subset \Omega$ which have a $C^1$ topological boundary $\partial E$ with $\mathcal{H}^{n-1} ((\partial E)\cap\Omega) < \infty$. If we denote by $\nu$ the exterior unit normal field at $\partial E$, the divergence theorem implies \begin{equation}\label{e:divergenza1} \int {\bf 1}_E\, {\rm div}\, \varphi\, d\lambda = - \int_E {\rm div}\, \varphi\, d\lambda = \int_{\partial E} \varphi\cdot \nu\, d\mathcal{H}^{n-1}\qquad \forall \varphi\in C^1_c (\Omega,\mathbb R^n)\, . \end{equation} Thus $V ({\bf 1}_E, \Omega) = \mathcal{H}^{n-1} ((\partial E)\cap\Omega)$, 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 \[ \inf \left\{ \liminf_k\; \mathcal{H}^{n-1} (\partial E_k):\;\{E_k\} \mbox{ is a sequence of smooth sets with } \lambda (E\bigtriangleup E_k) \to 0\right\}\, . \] 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]). 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 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]) and by $\partial_* E$ by others (see [EG]).

See Theorem 3.61 in [AFP]. In what follows we will stcik to the notation of [AFP] and use $\partial^* E$ for the essential boundary. The converse of Theorem 24 is also true, in the following sense: if $E\subset \Omega$ is a measurable set and $\mathcal{H}^{n-1} (K\setminus \{x: \theta^n (E,x)\in \{0,1\})< \infty$ for every compact subset $K\subset \Omega$, then $E$ is a Caccioppoli set. The latter is a deep theorem by Federer: see Theorem 4.5.11 of [Fe] and Section 5.11 of [EG]. Observe on the other hand that the following open subset of the real line \[ E := \bigcup_{k=1}^\infty ]2^{-k}, 2^{-k} + 3^{-k}[ \] is not a Caccioppoli set, although the density $\theta^1 (E,x)$ is either $0$, $\frac{1}{2}$ or $1$ at every point $x\in \mathbb R$.

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]).

Theorem 26 For any $x\in \mathcal{F} E$ the Lebesgue density $\theta^n (E,x)$ is equal to $\frac{1}{2}$, hence the reduced boundary is a subset of the essential boundary $\partial^* E$ and moreover $\mathcal{H}^{n-1} (\partial^* E\setminus\mathcal{F} E) = 0$. 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 can be generalized also to Caccioppoli sets. In particular the following fundamental result was first proved by De Giorgi, see [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 \[ \lambda (E) \leq C(n)\, \Big({\rm Per}\, (E, \mathbb R^n)\Big)^{\alpha}\, . \qquad \mbox{for any Caccioppoli set } E\subset\mathbb R^n \] and the equality holds if and only if $E$ is a ball.

A relative isoperimetric inequality holds also in extension domains $\Omega$, see Exercise 3.13 of [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 [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]. 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 [AFP]). A theorem due to Volpert (see [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]). The chain rule of Volpert has been generalized by Ambrosio and Dal Maso to Lipschitz $\varphi$ (see [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 ([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].

Special Functions of bounded variation

In [DA], in order to study variational problems involving free discontinuities (most notably the Mumford-Shah functional) De Giorgi and Ambrosio considered the 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, Ambrosio 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]), 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] for a comprehensive account of the theory of special functions of bounded variation.

Notable applications

Plateau's problem

Since their introduction 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] 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 [Br], [Da] and [Se].

Mumford-Shah functional

$SBV (\Omega)$ 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 direct methods of the calculus of variations.

Cahn-Hilliard

The 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].

References

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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=27775
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article