# Bounded variation-(only for study)

In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded (finite): the graph of a functionhaving this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed x-axis and to the y-axis.

Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions.

Another characterization states that the functions of bounded variation on a closed interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone.

In the case of several variables, a function f defined on an open subset $Omega$ of ℝn is said to have bounded variation if its distributional derivative is a finite vector Radon measure.

One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionalsordinary and partial differential equations in mathematicsphysics andengineering. Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations on generalized functionsfunction of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication.

[hide

## History

According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper (Jordan 1881) dealing with the convergence of Fourier series. The first successful step in the generalization of this concept to functions of several variables was due to Leonida Tonelli,[1] who introduced a class of continuous BV functions in 1926 (Cesari 1986, pp. 47–48), to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in (Cesari 1936), Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions oftwo variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theorycalculus of variations, and mathematical physics.Renato Caccioppoli and Ennio de Giorgi used them to define measure of nonsmooth boundaries of sets (see the entry "Caccioppoli set" for further information). Olga Arsenievna Oleinikintroduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper (Oleinik 1957), and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper (Oleinik 1959): few years later, Edward D. Conway and Joel A. Smoller applied BV-functions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper (Conway & Smoller 1966), proving that the solution of the Cauchy problemfor such equations is a function of bounded variation, provided the initial value belongs to the same class. Aizik Isaakovich Vol'pert developed extensively a calculus for BV functions: in the paper (Vol'pert 1967) he proved the chain rule for BV functions and in the book (Hudjaev & Vol'pert 1985) he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper (Ambrosio & Dal Maso 1990).

## Formal definition

### BV functions of one variable

Definition 1.1. The total variation[2] of a real-valued (or more generally complex-valued) function f, defined on an interval [ab]⊂ℝ is the quantity

$V^a_b(f)=sup_{P in mathcal{P}} sum_{i=0}^{n_{P}-1} | f(x_{i+1})-f(x_i) |. ,$

where the supremum is taken over the set $scriptstyle mathcal{P} =left{P={ x_0, dots , x_{n_P}}|Ptext{ is a partition of } [a, b] right}$ of all partitions of the interval considered.

If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,

$V^a_b(f) = int _a^b |f'(x)|,mathrm{d}x.$

Definition 1.2. A real-valued function $f$ on the real line is said to be of bounded variation (BV function) on a chosen interval [ab]⊂ℝ if its total variation is finite, i.e.

$f in BV([a,b]) iff V^a_b(f) < +infty$

It can be proved that a real function ƒ is of bounded variation in an interval if and only if it can be written as the difference ƒ = ƒ1 − ƒ2 of two non-decreasing functions: this result is known as the Jordan decomposition.

Through the Stieltjes integral, any function of bounded variation on a closed interval [ab] defines a bounded linear functional on C([ab]). In this special case,[3] the Riesz representation theorem states that every bounded linear functional arises uniquely in this way. The normalised positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in spectral theory,[4] in particular in its application to ordinary differential equations.

### BV functions of several variables

Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite[5] Radon measure. More precisely:

Definition 2.1. Let $Omega$ be an open subset of ℝn. A function $u$ belonging to $L^1(Omega)$ is said of bounded variation (BV function), and write

$uin BV(Omega)$

if there exists a finite vector Radon measure $scriptstyle Duinmathcal M(Omega,mathbb{R}^n)$ such that the following equality holds

$int_Omega u(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x = - int_Omega langleboldsymbol{phi}, Du(x)rangle qquad forallboldsymbol{phi}in C_c^1(Omega,mathbb{R}^n)$

that is, $u$ defines a linear functional on the space $scriptstyle C_c^1(Omega,mathbb{R}^n)$ of continuously differentiable vector functions $scriptstyleboldsymbol{phi}$ of compact support contained in $Omega$: the vector measure $Du$ represents therefore the distributional or weak gradient of $u$.

An equivalent definition is the following.

Definition 2.2. Given a function $u$ belonging to $L^1(Omega)$, the total variation of $u$[2] in $Omega$ is defined as

$V(u,Omega):=supleft{int_Omega u(x)mathrm{div}boldsymbol{phi}(x)mathrm{d}xcolonboldsymbol{phi}in C_c^1(Omega,mathbb{R}^n), Vertboldsymbol{phi}Vert_{L^infty(Omega)}le 1right}$

where $scriptstyle Vert;Vert_{L^infty(Omega)}$ is the essential supremum norm. Sometimes, especially in the theory of Caccioppoli sets, the following notation is used

$int_Omegavert D uvert = V(u,Omega)$

in order to emphasize that $V(u,Omega)$ is the total variation of the distributional / weak gradient of $u$. This notation reminds also that if $u$ is of class $C^1$ (i.e. a continuous anddifferentiable function having continuous derivatives) then its variation is exactly the integral of the absolute value of its gradient.

The space of functions of bounded variation (BV functions) can then be defined as

$BV(Omega)={ uin L^1(Omega)colon V(u,Omega)<+infty}$

The two definitions are equivalent since if $scriptstyle V(u,Omega)<+infty$ then

$left|int_Omega u(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x right |leq V(u,Omega)Vertboldsymbol{phi}Vert_{L^infty(Omega)} qquad forall boldsymbol{phi}in C_c^1(Omega,mathbb{R}^n)$

therefore $scriptstyle int_Omega u(x),mathrm{div}boldsymbol{phi}(x)$ defines a continuous linear functional on the space $scriptstyle C_c^1(Omega,mathbb{R}^n)$. Since $scriptstyle C_c^1(Omega,mathbb{R}^n) subset C^0(Omega,mathbb{R}^n)$ as a linear subspace, this continuous linear functional can be extendedcontinuously and linearily to the whole $scriptstyle C^0(Omega,mathbb{R}^n)$ by the Hahn–Banach theorem i.e. it defines a Radon measure.

### Locally BV functions

If the function space of locally integrable functions, i.e. functions belonging to $scriptstyle L^1_{loc}(Omega)$, is considered in the preceding definitions 1.22.1 and 2.2 instead of the one of globally integrable functions, then the function space defined is that of functions of locally bounded variation. Precisely, developing this idea for definition 2.2, a local variation is defined as follows,

$V(u,U):=supleft{int_Omega u(x)mathrm{div}boldsymbol{phi}(x)mathrm{d}xcolonboldsymbol{phi}in C_c^1(U,mathbb{R}^n), Vertboldsymbol{phi}Vert_{L^infty(Omega)}le 1right}$

for every set $scriptstyle Uinmathcal{O}_c(Omega)$, having defined $scriptstyle mathcal{O}_c(Omega)$ as the set of all precompact open subsets of $Omega$ with respect to the standard topology of finite dimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as

$BV_{loc}(Omega)={ uin L^1_{loc}(Omega)colon V(u,U)<+infty; forall Uinmathcal{O}_c(Omega)}$

### Notation

There are basically two distinct conventions for the notation of spaces of functions of locally or globally bounded variation, and unfortunately they are quite similar: the first one, which is the one adopted in this entry, is used for example in references Giusti (1984) (partially), Hudjaev & Vol'pert (1985) (partially), Giaquinta, Modica & Souček (1998) and is the following one

• $scriptstyle BV(Omega)$ identifies the space of functions of globally bounded variation
• $scriptstyle BV_{loc}(Omega)$ identifies the space of functions of locally bounded variation

The second one, which is adopted in references Vol'pert (1967) and Maz'ya (1985) (partially), is the following:

• $scriptstyle overline{BV}(Omega)$ identifies the space of functions of globally bounded variation
• $scriptstyle BV(Omega)$ identifies the space of functions of locally bounded variation

## Basic properties

Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and proofs will be carried on only for functions of several variables since the proof for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References (Giusti 1984, pp. 7–9), (Hudjaev & Vol'pert 1985) and (Màlek et al. 1996) are extensively used.

### BV functions have only jump-type discontinuities

In the case of one variable, the assertion is clear: for each point $x_0$ in the interval $[a , b]$⊂ℝ of definition of the function $u$, either one of the following two assertions is true

$lim_{xrightarrow x_{0^-}}!!!u(x) = !!!lim_{xrightarrow x_{0^+}}!!!u(x)$
$lim_{xrightarrow x_{0^-}}!!!u(x) neq !!!lim_{xrightarrow x_{0^+}}!!!u(x)$

while both limits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a continuum of directions along which it is possible to approach a given point $x_0$ belonging to the domain $Omega$⊂ℝn. It is necessary to make precise a suitable concept of limit: choosing a unit vector $scriptstyle{boldsymbol{hat{a}}}inmathbb{R}^n$ it is possible to divide $Omega$ in two sets

$Omega_{({boldsymbol{hat{a}}},boldsymbol{x}_0)} = Omega cap {boldsymbol{x}inmathbb{R}^n|langleboldsymbol{x}-boldsymbol{x}_0,{boldsymbol{hat{a}}}rangle>0} qquad Omega_{(-{boldsymbol{hat{a}}},boldsymbol{x}_0)} = Omega cap {boldsymbol{x}inmathbb{R}^n|langleboldsymbol{x}-boldsymbol{x}_0,-{boldsymbol{hat{a}}}rangle>0}$

Then for each point $x_0$ belonging to the domain $scriptstyleOmegainmathbb{R}^n$ of the BV function $u$, only one of the following two assertions is true

$lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{({boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!u(boldsymbol{x}) = !!!!!!!lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{(-{boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!!u(boldsymbol{x})$
$lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{({boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!u(boldsymbol{x}) neq !!!!!!!lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{(-{boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!!u(boldsymbol{x})$

or $x_0$ belongs to a subset of $Omega$ having zero $n-1$-dimensional Hausdorff measure. The quantities

$lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{({boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!u(boldsymbol{x})=u_{boldsymbol{hat a}}(boldsymbol{x}_0) qquad lim_{overset{boldsymbol{x}rightarrow boldsymbol{x}_0}{boldsymbol{x}inOmega_{(-{boldsymbol{hat{a}}},boldsymbol{x}_0)}}}!!!!!!!u(boldsymbol{x})=u_{-boldsymbol{hat a}}(boldsymbol{x}_0)$

are called approximate limits of the BV function $u$ at the point $x_0$.

### V(·, Ω) is lower semi-continuous on BV(Ω)

The functional $scriptstyle V(cdot,Omega):BV(Omega)rightarrow mathbb{R}^+$ is lower semi-continuous: to see this, choose a Cauchy sequence of BV-functions $scriptstyle{u_n}_{ninmathbb{N}}$ converging to $scriptstyle uin L^1_text{loc}(Omega)$. Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit

$liminf_{nrightarrowinfty}V(u_n,Omega) = liminf_{nrightarrowinfty} int_Omega u_n(x),mathrm{div}, boldsymbol{phi}, mathrm{d}x geq int_Omega lim_{nrightarrowinfty} u_n(x),mathrm{div}, boldsymbol{phi}, mathrm{d}x = int_Omega u(x),mathrm{div}boldsymbol{phi}, mathrm{d}x qquadforallboldsymbol{phi}in C_c^1(Omega,mathbb{R}^n),quadVertboldsymbol{phi}Vert_{L^infty(Omega)}leq 1$

Now considering the supremum on the set of functions $scriptstyleboldsymbol{phi}in C_c^1(Omega,mathbb{R}^n)$ such that $scriptstyle Vertboldsymbol{phi}Vert_{L^infty(Omega)}leq 1$ then the following inequality holds true

$liminf_{nrightarrowinfty}V(u_n,Omega)geq V(u,Omega)$

which is exactly the definition of lower semicontinuity.

### BV(Ω) is a Banach space

By definition $BV(Omega)$ is a subset of $L^1(Omega)$, while linearity follows from the linearity properties of the defining integral i.e.

begin{align} int_Omega [u(x)+v(x)],mathrm{div}boldsymbol{phi}(x)mathrm{d}x & = int_Omega u(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x +int_Omega v(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x = & =- int_Omega langleboldsymbol{phi}(x), Du(x)rangle- int_Omega langle boldsymbol{phi}(x), Dv(x)rangle =- int_Omega langle boldsymbol{phi}(x), [Du(x)+Dv(x)]rangle end{align}

for all $scriptstylephiin C_c^1(Omega,mathbb{R}^n)$ therefore $scriptstyle u+vin BV(Omega)$for all $scriptstyle u,vin BV(Omega)$, and

$int_Omega ccdot u(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x = c!int_Omega u(x),mathrm{div}boldsymbol{phi}(x)mathrm{d}x = -c! int_Omega langle boldsymbol{phi}(x), Du(x)rangle$

for all $scriptstyle cinmathbb{R}$, therefore $scriptstyle cuin BV(Omega)$ for all $scriptstyle uin BV(Omega)$, and all $scriptstyle cinmathbb{R}$. The proved vector space properties imply that $BV(Omega)$ is a vector subspace of $L^1(Omega)$. Consider now the function $scriptstyle|;|_{BV}:BV(Omega)rightarrowmathbb{R}^+$ defined as

$| u |_{BV} := | u |_{L^1} + V(u,Omega)$

where $scriptstyle| ; |_{L^1}$ is the usual $L^1(Omega)$ norm: it is easy to prove that this is a norm on $BV(Omega)$. To see that $BV(Omega)$ is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence $scriptstyle{u_n}_{ninmathbb{R}}$ in $BV(Omega)$. By definition it is also a Cauchy sequence in $L^1(Omega)$ and therefore has a limit $u$ in $L^1(Omega)$: since $u_n$ is bounded in $BV(Omega)$ for each $n$, then $scriptstyle Vert u Vert_{BV} < +infty$ by lower semicontinuity of the variation $scriptstyle V(cdot,Omega)$, therefore $u$ is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number $scriptstylevarepsilon$

$Vert u_j - u_k Vert_{BV}

### BV(Ω) is not separable

To see this, it is sufficient to consider the following example belonging to the space $BV([0,1])$[6]: for each 0<α<1 define

$chi_alpha=chi_{[alpha,1]}= begin{cases} 0 & mbox{if } x notin; [alpha,1] 1 & mbox{if } x in [alpha,1] end{cases}$

as the characteristic function of the left-closed interval $[alpha,1]$. Then, choosing α,β$[0,1]$ such that αβ the following relation holds true:

$Vert chi_alpha - chi_beta Vert_{BV}=2+|alpha-beta|$

Now, in order to prove that every dense subset of $BV(]0,1[)$ cannot be countable, it is sufficient to see that for every α$[0,1]$ it is possible to construct the balls

$B_alpha=left{psiin BV([0,1]);Vert chi_alpha - psi Vert_{BV}leq 1right}$

Obviously those balls are pairwise disjoint, and also are a indexed family of sets whose index set is $[0,1]$. This implies that this family has the cardinality of the continuum: now, since any dense subset of $BV([0,1])$ must have at least a point inside each member of this family, its cardinality is at least that of the continuum and therefore cannot a be countable subset.[7] This example can be obviously extended to higher dimensions, and since it involves only local properties, it implies that the same property is true for $BV_{loc}$.

### Chain rule for BV functions

Chain rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behavior is described by functionsor functionals with a very limited degree of smoothness.The following version is proved in the paper (Vol'pert 1967, p. 248): all partial derivatives must be intended in a generalized sense. i.e. as generalized derivatives

Theorem. Let $scriptstyle f:mathbb{R}^prightarrowmathbb{R}$ be a function of class $C^1$ (i.e. a continuous and differentiable function having continuous derivatives) and let $scriptstyleboldsymbol{u}(boldsymbol{x})=(u_1(boldsymbol{x}),ldots,u_p(boldsymbol{x}))$ be a function in $BV(Omega)$ with$Omega$ being an open subset of $scriptstylemathbb{R}^n$. Then $scriptstyle fcircboldsymbol{u}(boldsymbol{x})=f(boldsymbol{u}(boldsymbol{x}))in BV(Omega)$ and

$frac{partial f(boldsymbol{u}(boldsymbol{x}))}{partial x_i}=sum_{k=1}^pfrac{partialbar{f}(boldsymbol{u}(boldsymbol{x}))}{partial u_k}frac{partial{u_k(boldsymbol{x})}}{partial x_i} qquadforall i=1,ldots,n$

where $scriptstylebar f(boldsymbol{u}(boldsymbol{x}))$ is the mean value of the function at the point $scriptstyle x inOmega$, defined as

$bar f(boldsymbol{u}(boldsymbol{x}))=int_0^1 fleft(boldsymbol{u}_{boldsymbol{hat a}}(boldsymbol{x})t + boldsymbol{u}_{-boldsymbol{hat a}}(boldsymbol{x})(1-t)right)dt$

A more general chain rule formula for Lipschitz continuous functions $scriptstyle f:mathbb{R}^prightarrowmathbb{R}^s$ has been found by Luigi Ambrosio and Gianni Dal Maso and is published in the paper (Ambrosio & Dal Maso 1990). However, even this formula has very important direct consequences: choosing $scriptstyle f(u)=v(boldsymbol{x})u(boldsymbol{x})$, where $scriptstyle v(boldsymbol{x})$ is also a $BV$ function, the preceding formula gives the Leibniz rule for $BV$ functions

$frac{partial v(boldsymbol{x})u(boldsymbol{x})}{partial x_i} = {bar u(boldsymbol{x})}frac{partial v(boldsymbol{x})}{partial x_i} + {bar v(boldsymbol{x})}frac{partial u(boldsymbol{x})}{partial x_i}$

This implies that the product of two functions of bounded variation is again a function of bounded variation, therefore $BV(Omega)$ is an algebra.

### BV(Ω) is a Banach algebra

This property follows directly from the fact that $BV(Omega)$ is a Banach space and also an associative algebra: this implies that if ${v_n}$ and ${u_n}$ are Cauchy sequences of $BV$ functions converging respectively to functions $v$ and $u$ in $BV(Omega)$, then

$begin{matrix} vu_nxrightarrow[ntoinfty]{} vu v_nuxrightarrow[ntoinfty]{} vu end{matrix}quadLongleftrightarrow quad vuin BV(Omega)$

therefore the ordinary product of functions is continuous in $BV(Omega)$ respect to each argument, making this function space a Banach algebra.

## Generalizations and extensions

### Weighted BV functions

It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let $scriptstyle varphi : [0, +infty)longrightarrow [0, +infty)$ be any increasing function such that $scriptstyle varphi(0) = varphi(0+) =lim_{xrightarrow 0_+}varphi(x) = 0$ (the weight function) and let $scriptstyle f: [0, T]longrightarrow X$ be a function from the interval $[0 , T]$⊂ℝ taking values in a normed vector space $X$. Then the $scriptstyle boldsymbolvarphi$-variation of $f$ over $[0, T]$ is defined as

$mathop{varphimbox{-Var}}_{[0, T]} (f) := sup sum_{j = 0}^{k} varphi left( | f(t_{j + 1}) - f(t_{j}) |_{X} right),$

where, as usual, the supremum is taken over all finite partitions of the interval $[0, T]$, i.e. all the finite sets of real numbers $t_i$ such that

$0 = t_{0} < t_{1} < ldots < t_{k} = T.$

The original notion of variation considered above is the special case of $scriptstyle varphi$-variation for which the weight function is the identity function: therefore an integrable function $f$ is said to be a weighted BV function (of weight $scriptstylevarphi$) if and only if its $scriptstyle varphi$-variation is finite.

$fin BV_varphi([0, T];X)iff mathop{varphimbox{-Var}}_{[0, T]} (f) <+infty$

The space $scriptstyle BV_varphi([0, T];X)$ is a topological vector space with respect to the norm

$| f |_{BV_varphi} := | f |_{infty} + mathop{varphi mbox{-Var}}_{[0, T]} (f),$

where $scriptstyle| f |_{infty}$ denotes the usual supremum norm of $f$. Weighted BV functions were introduced and studied in full generality by Władysław Orlicz and Julian Musielak in the paper Musielak & Orlicz 1959Laurence Chisholm Young studied earlier the case $scriptstylevarphi(x)=x^p$ where $p$ is a positive integer.

### SBV functions

SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio de Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuityvariational problems: given an open subset $Omega$ of ℝn, the space $SBV(Omega)$ is a proper linear subspace of $BV(Omega)$, since the weak gradient of each function belonging to it consists precisely of the sum of an $n$-dimensional support and an $n-1$-dimensional support measure and no intermediate-dimensional terms, as seen in the following definition.

Definition. Given a locally integrable function $u$, then $scriptstyle uin {S!BV}(Omega)$ if and only if

1. There exist two Borel functions $f$ and $g$ of domain $Omega$ and codomain ℝn such that

$int_Omegavert fvert dH^n+ int_Omegavert gvert dH^{n-1}<+infty.$

2. For all of continuously differentiable vector functions $scriptstylephi$ of compact support contained in $Omega$i.e. for all $scriptstyle phi in C_c^1(Omega,mathbb{R}^n)$ the following formula is true:

$int_Omega umbox{div} phi dH^n = int_Omega langle phi, frangle dH^n +int_Omega langle phi, grangle dH^{n-1}.$

where $H^alpha$ is the $alpha$-dimensional Hausdorff measure.

Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper (De Giorgi 1992) contains a useful bibliography.

### bv sequences

As particular examples of Banach spacesDunford & Schwartz (1958, Chapter IV) consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x=(xi) of real or complex numbers is defined by

$TV(x) = sum_{i=1}^infty |x_{i+1}-x_i|.$

The space of all sequences of finite total variation is denoted by bv. The norm on bv is given by

$|x|_{bv} = |x_1| + TV(x) = |x_1| + sum_{i=1}^infty |x_{i+1}-x_i|.$

With this norm, the space bv is a Banach space.

The total variation itself defines a norm on a certain subspace of bv, denoted by bv0, consisting of sequences x = (xi) for which

$lim_{ntoinfty} x_n =0.$

The norm on bv0 is denoted

$|x|_{bv_0} = TV(x) = sum_{i=1}^infty |x_{i+1}-x_i|.$

With respect to this norm bv0 becomes a Banach space as well.

### Measures of bounded variation

signed (or complexmeasure $mu$ on a measurable space $(X,Sigma)$ is said to be of bounded variation if its total variation $scriptstyleVert muVert=|mu|(X)$ is bounded: see Halmos (1950, p. 123), Kolmogorov & Fomin (1969, p. 346) or the entry "Total variation" for further details.

## Examples

The function f(x)=sin(1/x) is notof bounded variation on the interval$[0,2 / pi]$.

The function

$f(x) = begin{cases} 0, & mbox{if }x =0 sin(1/x), & mbox{if } x neq 0 end{cases}$

is not of bounded variation on the interval $[0, 2/pi]$

The function f(x)=x sin(1/x) isnot of bounded variation on the interval $[0,2 / pi]$.

While it is harder to see, the continuous function

$f(x) = begin{cases} 0, & mbox{if }x =0 x sin(1/x), & mbox{if } x neq 0 end{cases}$

is not of bounded variation on the interval $[0, 2/pi]$ either.

The function f(x)=x2 sin(1/xisof bounded variation on the interval$[0,2 / pi]$.

At the same time, the function

$f(x) = begin{cases} 0, & mbox{if }x =0 x^2 sin(1/x), & mbox{if } x neq 0 end{cases}$

is of bounded variation on the interval $[0,2/pi]$. However, all three functions are of bounded variation on each interval $[a,b]$ with $a>0$.

The Sobolev space $W^{1,1}(Omega)$ is a proper subset of $BV(Omega)$. In fact, for each $u$ in $W^{1,1}(Omega)$ it is possible to choose a measure $scriptstyle mu:=nabla u mathcal L$ (where $scriptstylemathcal L$ is the Lebesgue measure on $Omega$) such that the equality

$int umathrm{div}phi = -int phi, dmu = -int phi nabla u qquad forall phiin C_c^1$

holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a BV function which is not $W^{1,1}$: in dimension one, any step function with a non-trivial jump will do.

## Applications

### Mathematics

Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. If $f$ is a real function of bounded variation on an interval $[a,b]$ then

• $f$ is continuous except at most on a countable set;
• $f$ has one-sided limits everywhere (limits from the left everywhere in $(a,b]$, and from the right everywhere in $[a,b)$ ;
• the derivative $f'(x)$ exists almost everywhere (i.e. except for a set of measure zero).

For real functions of several real variables

### Physics and engineering

The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation. The book (Hudjaev & Vol'pert 1985) details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.

## Notes

1. ^ Tonelli introduced what is now called after him Tonelli plane variation: for an analysis of this concept and its relations to other generalizations, see the entry "Total variation".
2. a b See the entry "Total variation" for further details and more information.
3. ^ See for example Kolmogorov & Fomin (1969, pp. 374–376).
4. ^ For a general reference on this topic, see Riesz & Szőkefalvi-Nagy (1990)
5. ^ In this context, "finite" means that its value is neverinfinite, i.e. it is a finite measure.
6. ^ The example is taken from Giaquinta, Modica & Souček (1998, p. 331)
7. ^ The reasoning is the same used by Kolmogorov & Fomin 1969, pp. 48–49, example 7, in order to prove the non separability of the space of bounded sequences.

## 

View page ratings

Trustworthy

Objective

Complete

Well-written