Function of bounded variation
2010 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)
2010 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)
Contents
- 1 Functions of one variable
- 2 Functions of several variables
- 3 Slicing
- 4 Caccioppoli sets
- 5 Volpert chain rule
- 6 Alberti's rank-one theorem
- 7 Special Functions of bounded variation
- 8 Notable applications
- 9 References
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}\tag{1} 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 (1). 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 patologies as in Warning 6 it is customary to postulate some additional assumptions for functions of bounded variations. Two popular choices are
- the imposition of right (resp. left) continuity, i.e. at any point $x$ we impose $f(x)=f (x^+)$ (resp. $f(x)=f(x^-$), cp. with Section 4.4 of [Co];
- at any point $x$ we impose $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.
The latter is perhaps more popular because of the Jordan criterion (see Theorem 11 below) and it is often called precise representative.
Differentiability
Functions of bounded variation of one variable are classically differentiable at a.e. point of their domain of definition, cp. with Corollary 5 of Section 5.2 in [Ro]. It turns out that such derivative is always a summable function (see below in the section Structure theorem). However, the fundamental theorem of calculus does not apply in this case, i.e. there are continuous functions $f:[a,b]\to\mathbb R$ of bounded variation such that the identity \[ f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt \] fails for a set of pairs $(b', a')\in I\times I$ of positive measure (see below in the section Examples).
Measure theoretic characterization
Classically right-continuous functions of bounded variations can be mapped one-to-one to signed measures. More precisely, consider a signed measure $\mu$ on (the Borel subsets of) $\mathbb R$ with finite total variation (see Signed measure for the definition). We then define the function \begin{equation}\tag{2} 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}\tag{3} \int f(t)\varphi' (t)\, dt = -\int \varphi (t)\, d\mu (t) \qquad \forall \varphi\in C^\infty_c (\mathbb R)\, . \end{equation} This identty is the starting point for the modern definition of functions of bounded variation, cp. with [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}\tag{4} \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. (3) 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 (4).
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
- $f_a$ is an absolutely continuous function
- $f_c$ is a singular function
- $f_j$ is a 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.
Examples
Smooth functions
If $f: I\to\mathbb R$ is smooth, then we have the identity \begin{equation}\tag{5} 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 (5) 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 (5) 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 (5) 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}\tag{6} \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}\tag{7} 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}\tag{8} 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 (8) 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}\tag{9} \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 (9) 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}\tag{10} \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 \[ \mu (A) := - \int_{A\cap E} \nu\, d\mathcal{H}^{n-1}\, , \] (10) 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}\tag{11} \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 (11) 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 and it is a deep theorem by Federer: see Section 5.11 of [EG].
Reduced boundary and structure theorem
The essential boundary of a Caccioppoli set can be analyzed further.
Definition 25 If $E\subset\Omega$ is a Caccioppoli set the reduced boundary of $E$ is defined as \[ \mathcal{F} E := \left\{ x\in\Omega : \nu_E (x) := \lim_{r\downarrow 0} \frac{D{\bf 1}_E (B_r(x))}{|D {\bf 1}_E| (B_r(x))}\;\; \mbox{exists and } |\nu_E (x)|=1 \right\}\, . \] $\nu_E$ is called the measure theoretic inner normal.
We then have the following fundamental result, due to De Giorgi (for a proof see Section 3.5 of [AFP]).
Theorem 26 For any $x\in \mathcal{F} E$ the Lebesgue density $\theta^n (E,x)$ is equal to $\frac{1}{2}$ and hence the reduced boundary is a subset of the essential boundary (and, by Theorem 23, $\mathcal{H}^{n-1} (\partial^* E\setminus\mathcal{F} E) = 0$). The set $\mathcal{F} E$ is a rectifiable set and $\nu_E$ is orthogonal to it $\mathcal{H}^{n-1}$-a.e.. Finally we have the identity \begin{equation}\tag{12} 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}\tag{13} \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}\tag{14} \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].
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