# Bounded variation

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} 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 function having 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 functionals, ordinary and partial differential equations in mathematics, physics and engineering. Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations on generalized functions, functions of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication.

## History

According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper Template:Harv 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, who introduced a class of continuous BV functions in 1926 Template:Harv, to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in Template:Harv, 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 of two variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus 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 Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper Template:Harv, and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper Template:Harv: 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 Template:Harv, proving that the solution of the Cauchy problem for 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 Template:Harv he proved the chain rule for BV functions and in the book Template:Harv 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 Template:Harv.

## Formal definition

### BV functions of one variable

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

$V_{a}^{b}(f)=\sup _{P\in {\mathcal {P}}}\sum _{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|.\,$ 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 [a, b]⊂ℝ if its total variation is finite, i.e.

$f\in BV([a,b])\iff V_{b}^{a}(f)<+\infty$ 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 = ƒ1ƒ2 of two non-decreasing functions: this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure.

Through the Stieltjes integral, any function of bounded variation on a closed interval [a, b] defines a bounded linear functional on C([a, b]). In this special case, 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, 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 Radon measure. More precisely:

$u\in BV(\Omega )$ $\int _{\Omega }u(x)\,{\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x=-\int _{\Omega }\langle {\boldsymbol {\phi }},Du(x)\rangle \qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,{\mathbb {R} }^{n})$ An equivalent definition is the following.

$V(u,\Omega ):=\sup \left\{\int _{\Omega }u(x){\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x\colon {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,{\mathbb {R} }^{n}),\ \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1\right\}$ where $\Vert \;\Vert _{L^{\infty }(\Omega )}$ is the essential supremum norm. Sometimes, especially in the theory of Caccioppoli sets, the following notation is used

$\int _{\Omega }\vert Du\vert =V(u,\Omega )$ The space of functions of bounded variation (BV functions) can then be defined as

$BV(\Omega )=\{u\in L^{1}(\Omega )\colon V(u,\Omega )<+\infty \}$ The two definitions are equivalent since if $V(u,\Omega )<+\infty$ then

$\left|\int _{\Omega }u(x)\,{\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x\right|\leq V(u,\Omega )\Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,{\mathbb {R} }^{n})$ ### Locally BV functions

If the function space of locally integrable functions, i.e. functions belonging to $L_{loc}^{1}(\Omega )$ , is considered in the preceding definitions Template:EquationNote, Template:EquationNote and Template:EquationNote 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 Template:EquationNote, a local variation is defined as follows,

$V(u,U):=\sup \left\{\int _{\Omega }u(x){\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x\colon {\boldsymbol {\phi }}\in C_{c}^{1}(U,{\mathbb {R} }^{n}),\ \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1\right\}$ for every set $U\in {\mathcal {O}}_{c}(\Omega )$ , having defined ${\mathcal {O}}_{c}(\Omega )$ 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

$BV_{loc}(\Omega )=\{u\in L_{loc}^{1}(\Omega )\colon V(u,U)<+\infty \;\forall U\in {\mathcal {O}}_{c}(\Omega )\}$ ### 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 Template:Harvtxt (partially), Template:Harvtxt (partially), Template:Harvtxt and is the following one

The second one, which is adopted in references Template:Harvtxt and Template:Harvtxt (partially), is the following:

## 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 Template:Harv, Template:Harv and Template:Harv 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

$\lim _{x\rightarrow x_{0^{-}}}\!\!\!u(x)=\!\!\!\lim _{x\rightarrow x_{0^{+}}}\!\!\!u(x)$ $\lim _{x\rightarrow x_{0^{-}}}\!\!\!u(x)\neq \!\!\!\lim _{x\rightarrow x_{0^{+}}}\!\!\!u(x)$ 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 ${\boldsymbol {\hat {a}}}\in {\mathbb {R} }^{n}$ it is possible to divide $\Omega$ in two sets

$\Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}=\Omega \cap \{{\boldsymbol {x}}\in {\mathbb {R} }^{n}|\langle {\boldsymbol {x}}-{\boldsymbol {x}}_{0},{\boldsymbol {\hat {a}}}\rangle >0\}\qquad \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}=\Omega \cap \{{\boldsymbol {x}}\in {\mathbb {R} }^{n}|\langle {\boldsymbol {x}}-{\boldsymbol {x}}_{0},-{\boldsymbol {\hat {a}}}\rangle >0\}$ $\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})=\!\!\!\!\!\!\!\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})$ $\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})\neq \!\!\!\!\!\!\!\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})$ $\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})=u_{\boldsymbol {\hat {a}}}({\boldsymbol {x}}_{0})\qquad \lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})=u_{-{\boldsymbol {\hat {a}}}}({\boldsymbol {x}}_{0})$ ### V(·, Ω) is lower semi-continuous on BV(Ω)

$\liminf _{n\rightarrow \infty }V(u_{n},\Omega )\geq \liminf _{n\rightarrow \infty }\int _{\Omega }u_{n}(x)\,\mathrm {div} \,{\boldsymbol {\phi }}\,\mathrm {d} x\geq \int _{\Omega }\lim _{n\rightarrow \infty }u_{n}(x)\,\mathrm {div} \,{\boldsymbol {\phi }}\,\mathrm {d} x=\int _{\Omega }u(x)\,\mathrm {div} {\boldsymbol {\phi }}\,\mathrm {d} x\qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\quad \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1$ $\liminf _{n\rightarrow \infty }V(u_{n},\Omega )\geq V(u,\Omega )$ which is exactly the definition of lower semicontinuity.

### BV(Ω) is a Banach space

{\begin{aligned}\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 }\langle {\boldsymbol {\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{aligned}} $\int _{\Omega }c\cdot u(x)\,{\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x=c\!\int _{\Omega }u(x)\,{\mathrm {div} }{\boldsymbol {\phi }}(x){\mathrm {d} }x=-c\!\int _{\Omega }\langle {\boldsymbol {\phi }}(x),Du(x)\rangle$ $\|u\|_{BV}:=\|u\|_{L^{1}}+V(u,\Omega )$ $\Vert u_{j}-u_{k}\Vert _{BV}<\varepsilon \quad \forall j,k\geq N\in {\mathbb {N} }\quad \Rightarrow \quad V(u_{k}-u,\Omega )\leq \liminf _{j\rightarrow +\infty }V(u_{k}-u_{j},\Omega )\leq \varepsilon$ ### BV(Ω) is not separable

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

$\chi _{\alpha }=\chi _{[\alpha ,1]}={\begin{cases}0&{\mbox{if }}x\notin \;[\alpha ,1]\\1&{\mbox{if }}x\in [\alpha ,1]\end{cases}}$ $\Vert \chi _{\alpha }-\chi _{\beta }\Vert _{BV}=2+|\alpha -\beta |$ 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

$B_{\alpha }=\left\{\psi \in BV([0,1]);\Vert \chi _{\alpha }-\psi \Vert _{BV}\leq 1\right\}$ Obviously those balls are pairwise disjoint, and also are an 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. This example can be obviously extended to higher dimensions, and since it involves only local properties, it implies that the same property is true also 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 functions or functionals with a very limited degree of smoothness.The following version is proved in the paper Template:Harv: all partial derivatives must be intended in a generalized sense. i.e. as generalized derivatives

${\frac {\partial f({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial x_{i}}}=\sum _{k=1}^{p}{\frac {\partial {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial u_{k}}}{\frac {\partial {u_{k}({\boldsymbol {x}})}}{\partial x_{i}}}\qquad \forall i=1,\ldots ,n$ ${\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))=\int _{0}^{1}f\left({\boldsymbol {u}}_{\boldsymbol {\hat {a}}}({\boldsymbol {x}})t+{\boldsymbol {u}}_{-{\boldsymbol {\hat {a}}}}({\boldsymbol {x}})(1-t)\right)dt$ ${\frac {\partial v({\boldsymbol {x}})u({\boldsymbol {x}})}{\partial x_{i}}}={{\bar {u}}({\boldsymbol {x}})}{\frac {\partial v({\boldsymbol {x}})}{\partial x_{i}}}+{{\bar {v}}({\boldsymbol {x}})}{\frac {\partial u({\boldsymbol {x}})}{\partial x_{i}}}$ 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

${\begin{matrix}vu_{n}{\xrightarrow[{n\to \infty }]{}}vu\\v_{n}u{\xrightarrow[{n\to \infty }]{}}vu\end{matrix}}\quad \Longleftrightarrow \quad vu\in BV(\Omega )$ 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

${\mathop {\varphi {\mbox{-Var}}}}_{[0,T]}(f):=\sup \sum _{j=0}^{k}\varphi \left(|f(t_{j+1})-f(t_{j})|_{X}\right),$ 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} The original notion of variation considered above is the special case of $\varphi$ -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 $\varphi$ ) if and only if its $\varphi$ -variation is finite.

$f\in BV_{\varphi }([0,T];X)\iff {\mathop {\varphi {\mbox{-Var}}}}_{[0,T]}(f)<+\infty$ $\|f\|_{BV_{\varphi }}:=\|f\|_{\infty }+{\mathop {\varphi {\mbox{-Var}}}}_{[0,T]}(f),$ ### SBV functions

SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio de Giorgi in the paper Template:Harv, dealing with free discontinuity variational 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.

$\int _{\Omega }\vert f\vert dH^{n}+\int _{\Omega }\vert g\vert dH^{n-1}<+\infty .$ $\int _{\Omega }u{\mbox{div}}\phi dH^{n}=\int _{\Omega }\langle \phi ,f\rangle dH^{n}+\int _{\Omega }\langle \phi ,g\rangle dH^{n-1}.$ Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper Template:Harv contains a useful bibliography.

### bv sequences

As particular examples of Banach spaces, Template:Harvtxt 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

$\lim _{n\to \infty }x_{n}=0.$ 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.

## Examples

The function

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

At the same time, the function

$f(x)={\begin{cases}0,&{\mbox{if }}x=0\\x^{2}\sin(1/x),&{\mbox{if }}x\neq 0\end{cases}}$ $\int u{\mathrm {div} }\phi =-\int \phi \,d\mu =-\int \phi \nabla u\qquad \forall \phi \in C_{c}^{1}$ 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

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 Template:Harv details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.