- Bounded variation
In

mathematical analysis , a function of**bounded variation**refers to a real-valued function whosetotal variation is bounded (finite): thegraph of a function having this property is well behaved in a precise sense. For acontinuous function of a singlevariable , being of bounded variation means that thedistance along the direction ofy-axis (i.e. the distance calculated neglecting the contribution of motion alongx-axis ) traveled by an ideal point moving along the graph of the given function (which, under given hypothesis, is also a continuous path) 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 an hypersurface in this case), but can be every intersection of the graph itself with 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 integral s 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 $scriptstylemathbb\{R\}^n$ is said to have bounded variation if its distributional derivative is a finite vectorRadon 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 definegeneralized solution s of nonlinear problems involving functionals, ordinary andpartial differential equation s inmathematics ,physics andengineering . Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations ongeneralized function s, "function of bounded variation are the smallest algebra which has to be embedded in every space ofgeneralized function s preserving the result ofmultiplication ".**History**According to Golubov, "BV" functions of a single variable were first introduced by

Camille Jordan , in the paper Harv|Jordan|1881 dealing with the convergence ofFourier series . The first step in the generalization of this concept to functions of several variables was due toLeonida Tonelli , who introduced a class of "continuous" "BV" functions in 1926 Harv|Cesari|1986|pp=47-48, to extend his direct method for finding solutions to problems in thecalculus of variations in more than one variable. Ten years after, in 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 ofFourier series , but for functions of "two variables". After him, several authors applied "BV" functions to studyFourier series in several variables,geometric measure theory ,calculus of variations , andmathematical physics :Renato Caccioppoli andEnnio de Giorgi used them to define measure of non smooth boundaries of sets (see voice "Caccioppoli set " for further informations),Edward D. Conway andJoel A. Smoller applied them to the study of a single nonlinear hyperbolic partial differential equation offirst order in the paper Harv|Conway|Smoller|1966, proving that the solution of theCauchy 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 Harv|Vol'pert|1967 and in the book Harv|Hudjaev|Vol'pert|1986.**Formal definition****"BV" functions of one variable****Definition 1.**The**total variation**of a real-valued function "$f$", defined on an interval $scriptstyle\; [a\; ,\; b]\; subset\; mathbb\{R\}$ 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 $scriptstyle\; mathcal\{P\}\; =left\{P=\{\; x\_0,\; dots\; ,\; x\_\{n\_p\}\}|P\; ext\{\; is\; a\; partition\; of\; \}\; [a,b]\; ight\}$ of all partitions of the interval considered.If $f$ is differentiable and its derivative is integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,

:$V^a\_b(f)\; =\; int\; \_a^b\; |f\text{'}(x)|,\; dx.$

**Definition 2**. A real-valued function $f$ on thereal line is said to be of**bounded variation**(**BV function**) on a chosen interval $[a,b]$ 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 f is of bounded variation in an interval if and only if it can be written as the difference $f=f\_1\; -\; f\_2$ of two non decreasing functions (This is known as the

Jordan decomposition .)Through the

Stieltjes integral , any function of bounded variation on a closed interval $[a,b]$ defines abounded linear functional on $C(\; [a,b]\; )$. In this special case harv|Kolmogorov|Fomin|1969|pp=374-376, theRiesz representation theorem states that every bounded linear functional arises uniquely in this way. The normalised positive functionals orprobability measure s correspond to positive non-decreasing lowersemicontinuous function s. This point of view has been important inspectral theory harv|Riesz|Sz.-Nagy|1990, 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:**Definition 1**Let**$Omega$**be anopen subset of $scriptstylemathbb\{R\}^n$. Alocally integrable function **$u$**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:$int\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x),\; dx\; =\; -\; int\_Omega\; langle\backslash boldsymbol\{phi\},\; Du(x)\; angle\; qquad\; forall\backslash boldsymbol\{phi\}in\; C\_c^1(Omega,mathbb\{R\}^n)$

that is,

**$u$**defines alinear functional on the space $scriptstyle\; C\_c^1(Omega,mathbb\{R\}^n)$ of continuously differentiable vector functions $scriptstyle\backslash boldsymbol\{phi\}$ of compact support contained in**$Omega$**: the vector measure**$Du$**represents therefore the distributional or weakgradient of**$u$**.An equivalent definition is the following.

**Definition 2**Given alocally integrable function **$u$**, the**total variation of $u$**in is defined as:$V(u,Omega):=supleft\{int\_Omega\; umathrm\{div\}\backslash boldsymbol\{phi\}colon\; phiin\; C\_c^1(Omega,mathbb\{R\}^n),\; Vert\backslash boldsymbol\{phi\}Vert\_\{L^infty(Omega)\}le\; 1\; ight\}.$

where $scriptstyle\; Vert;Vert\_\{L^infty(Omega)\}$ is the

essential supremum norm.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 definition are equivalent**since if $scriptstyle\; V(u,Omega)<+infty$ then:$left|int\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\},\; dx\; ight\; |leq\; V(u,Omega)Vert\backslash boldsymbol\{phi\}Vert\_\{L^infty(Omega)\}qquad\; forall\; \backslash boldsymbol\{phi\}in\; C\_c^1(Omega,mathbb\{R\}^n)$

therefore $scriptstyle\; int\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x)$ 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 alinear subspace , thiscontinuous linear functional can be extended continuously and linearily to the whole $scriptstyle\; C^0(Omega,mathbb\{R\}^n)$ by theHahn–Banach theorem i.e. it defines a Radon measure.**Basic properties**Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and

proof s will be carried on only for functions of several variables since theproof for the case of one variable is a straightforward adaptation of the several variables case. References Harv|Giusti|1984|pp=7-9, Harv|Hudjaev|Vol'pert|1986 and Harv|Màlek|Nečas|Rokyta|Růžička|1996 are extensively used.**"BV" functions have only jump-type singularities**In the case of one variable, the assertion is clear: for each point

**$x\_0$**in the interval $scriptstyle\; ]\; a\; ,\; b\; [\; subset\; mathbb\{R\}$ of definition of the function**$u$**, either one of the following two assertions is true:$lim\_\{x\; ightarrow\; x\_\{0^-!!!u(x)\; =\; !!!lim\_\{x\; ightarrow\; x\_\{0^+!!!u(x)$:$lim\_\{x\; ightarrow\; x\_\{0^-!!!u(x)\; eq\; !!!lim\_\{x\; ightarrow\; x\_\{0^+!!!u(x)$

while both limits exists and are finite. In the case of functions 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 $scriptstyleOmegainmathbb\{R\}^n$. It is necessary to make precise a suitable concept of limit: choosing aunit vector $scriptstyle\{oldsymbolhat\; a\}inmathbb\{R\}^n$ it is possible to divide**$Omega$**in two sets:$Omega\_\{(\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}\; =\; Omega\; cap\; \{\backslash boldsymbol\{x\}inmathbb\{R\}^n|langle\backslash boldsymbol\{x\}-\backslash boldsymbol\{x\}\_0,\{\backslash boldsymbolhat\; a\}\; angle0\}\; qquad\; Omega\_\{(-\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}\; =\; Omega\; cap\; \{\backslash boldsymbol\{x\}inmathbb\{R\}^n|langle\backslash boldsymbol\{x\}-\backslash boldsymbol\{x\}\_0,-\{\backslash boldsymbolhat\; a\}\; angle0\}$

Then for each point

**$x\_0$**belonging to the domain $scriptstyleOmegainmathbb\{R\}^n$ of the "BV" function**$u$**or one of the following two assertion is true:$lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!u(\backslash boldsymbol\{x\})\; =\; !!!!!!!lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(-\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!!u(\backslash boldsymbol\{x\})$:$lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!u(\backslash boldsymbol\{x\})\; eq\; !!!!!!!lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(-\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!!u(\backslash boldsymbol\{x\})$

or

**$x\_0$**belongs to asubset of**$Omega$**having zero $n-1$-dimensional Hausdorff measure. The quantities:$lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!u(\backslash boldsymbol\{x\})=u\_\{\backslash boldsymbol\{hat\; a(\backslash boldsymbol\{x\})\; qquad\; lim\_\{overset\{\backslash boldsymbol\{x\}\; ightarrow\; \backslash boldsymbol\{x\}\_0\}\{\backslash boldsymbol\{x\}inOmega\_\{(-\{\backslash boldsymbolhat\; a\},\backslash boldsymbol\{x\}\_0)\}!!!!!!!u(\backslash boldsymbol\{x\})=u\_\{-\backslash boldsymbol\{hat\; a(\backslash boldsymbol\{x\})$

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 $scriptstyle\; V(cdot,Omega):BV(Omega)\; ightarrow\; mathbb\{R\}^+$ is lower semi-continuous: to see this, choose aCauchy sequence of "BV"-functions**$scriptstyle\{u\_n\}\_\{ninmathbb\{N$**converging to**$scriptstyle\; uin\; L^1\_\{loc\}(Omega)$**. Then, since all the functions of the sequence and their limit function are integrable and by the definition oflower limit :$liminf\_\{n\; ightarrowinfty\}V(u\_n,Omega)geqlim\_\{n\; ightarrowinfty\}int\_Omega\; u\_n(x),mathrm\{div\}\backslash boldsymbol\{phi\},\; dx\; =\; int\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\},\; dx\; qquadforall\backslash boldsymbol\{phi\}in\; C\_c^1(Omega,mathbb\{R\}^n),quadVert\backslash boldsymbol\{phi\}Vert\_\{L^infty(Omega)\}leq\; 1$

Now considering the

supremum on the set of functions $scriptstyle\backslash boldsymbol\{phi\}in\; C\_c^1(Omega,mathbb\{R\}^n)$ such that $scriptstyle\; Vert\backslash boldsymbol\{phi\}Vert\_\{L^infty(Omega)\}leq\; 1$ then the following inequality holds true:$liminf\_\{n\; ightarrowinfty\}V(u\_n,Omega)geq\; V(u,Omega)$

which is exactly the definition of lower semicontinuity.

**BV(Ω) is a Banach space**By definition

**$BV(Omega)$**is asubset of**"L"**, while^{"1"}_{"loc"}(Ω)linearity follows from the linearity properties of the definingintegral i.e.:$egin\{align\}int\_Omega\; [u(x)+v(x)]\; ,mathrm\{div\}\backslash boldsymbol\{phi\}(x)\; =int\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x)+int\_Omega\; v(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x)\; =\; \backslash \; =-\; int\_Omega\; langle\backslash boldsymbol\{phi\}(x),\; Du(x)\; angle-\; int\_Omega\; langle\; \backslash boldsymbol\{phi(x)\},\; Dv(x)\; angle\; =-\; int\_Omega\; langle\; \backslash boldsymbol\{phi\}(x),\; [Du(x)+Dv(x)]\; angle\; end\{align\}$

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

:$int\_Omega\; cu(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x)=cint\_Omega\; u(x),mathrm\{div\}\backslash boldsymbol\{phi\}(x)=-c\; int\_Omega\; langle\; \backslash boldsymbol\{phi\}(x),\; Du(x)\; angle$

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 avector subspace of**$L^1(Omega)$**. Consider now the function $scriptstyle|;|\_\{BV\}:BV(Omega)\; ightarrowmathbb\{R\}^+$ 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 anorm on**$BV(Omega)$**. To see that**$BV(Omega)$**is complete respect to it, i.e. it is aBanach space , consider aCauchy sequence $scriptstyle\{u\_n\}\_\{ninmathbb\{R$ in**$BV(Omega)$**. By definition it is also aCauchy sequence in**$L^1(Omega)$**and therefore has alimit **$u$**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**$scriptstylevarepsilon$**:$Vert\; u\_j\; -\; u\_k\; Vert\_\{BV\},kgeq\; ninmathbb\{n\}\; quadrightarrowquad\; v(u\_k-u,omega)leq\; liminf\_\{j\; ightarrow\; +infty\}\; v(u\_k-u\_j,omega)leqvarepsilon\; math>$

**Chain rule for "BV" functions**Chain rule s for nonsmooth functions are very important inmathematics andmathematical physics since there are several importantphysical model s whose behavior is described by functions orfunctional s with a very limited degree of smoothness.The following version is proved in the paper Harv|Vol'pert|1967|p=248: allpartial derivative s must be intended in a generalized sense. i.e. as generalized derivatives**Theorem**. Let $scriptstyle\; f:mathbb\{R\}^p\; ightarrowmathbb\{R\}$ be a function of class**$C^1$**(i.e. a continuous anddifferentiable function having continuousderivative s) and let $scriptstyle\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\})=(u\_1(\backslash boldsymbol\{x\}),ldots,u\_p(\backslash boldsymbol\{x\}))$ be a function in**$BV(Omega)$**with**$Omega$**being anopen subset of $scriptstylemathbb\{R\}^n$.Then $scriptstyle\; fcirc\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\})=f(\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\}))in\; BV(Omega)$ and:$frac\{partial\; f(\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\}))\}\{partial\; x\_i\}=sum\_\{k=1\}^pfrac\{partialar\{f\}(\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\}))\}\{partial\; u\_k\}frac\{partial\{u\_k(\backslash boldsymbol\{x\})\{partial\; x\_i\}qquadforall\; i=1,ldots,n$

where $scriptstylear\; f(\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\}))$ is the mean value of the function at the point

**$scriptstyle\; x\; inOmega$**, defined as:$ar\; f(\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\}))=int\_0^1\; fleft(\backslash boldsymbol\{u\}\_\{\backslash boldsymbol\{hat\; a(\backslash boldsymbol\{x\})t\; +\; \backslash boldsymbol\{u\}\_\{-\backslash boldsymbol\{hat\; a(\backslash boldsymbol\{x\})(1-t)\; ight)dt$

A more general

chain rule formula for Lipschitz continuous functions $scriptstyle\; f:mathbb\{R\}^p\; ightarrowmathbb\{R\}^s$ has been found byLuigi Ambrosio andGianni Dal Maso and published in the paper Harv|Ambrosio|Dal Maso|1990. However, even this formula has very important direct consequences: choosing $scriptstyle\; f(u)=v(\backslash boldsymbol\{x\})u(\backslash boldsymbol\{x\})$ where $scriptstyle\; v(\backslash boldsymbol\{x\})$ is a "BV" function the preceding formula becomes the**"Leibnitz rule**" for**$BV$**functions:$frac\{partial\; v(\backslash boldsymbol\{x\})u(\backslash boldsymbol\{x\})\}\{partial\; x\_i\}\; =\; \{ar\; u(\backslash boldsymbol\{x\})\}frac\{partial\; v(\backslash boldsymbol\{x\})\}\{partial\; x\_i\}\; +\; \{ar\; v(\backslash boldsymbol\{x\})\}frac\{partial\; u(\backslash boldsymbol\{x\})\}\{partial\; x\_i\}$

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 aBanach space and also anassociative algebra : this implies that if**$\{v\_n\}$**and**$\{u\_n\}$**areCauchy sequence s of $BV$ functions converging respectively to functions**$v$**and**$u$**in**$BV(Omega)$**, then::$egin\{matrix\}\; vu\_nxrightarrow\; [n\; oinfty]\; \{\}\; vu\; \backslash \; v\_nuxrightarrow\; [n\; oinfty]\; \{\}\; vu\; end\{matrix\}qquad\; vuin\; BV(Omega)$

therefore the ordinary product of functions is continuous in

**$BV(Omega)$**respect to each argument, making this function space aBanach 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 $scriptstyle\; varphi(0)\; =\; varphi(0+)\; =lim\_\{x\; ightarrow\; 0\_+\}varphi(x)\; =\; 0$ (the) and let $scriptstyle\; f:\; [0,\; T]\; longrightarrow\; X$ be a function from the interval $scriptstyle\; [0\; ,\; T]\; subset\; mathbb\{R\}$ taking values in aweight function normed vector space $X$. Then the $scriptstyle\; oldsymbolvarphi$**-variation**of $f$ over $[0,\; T]$ is defined as:$mathop\{varphimbox\{-Var\_\{\; [0,\; T]\; \}\; (f)\; :=\; sup\; sum\_\{j\; =\; 0\}^\{k\}\; varphi\; left(\; |\; f(t\_\{j\; +\; 1\})\; -\; f(t\_\{j\})\; |\_\{X\}\; ight),$

where, as usual, the supremum is taken over all finite partitions of the interval $[0,\; T]$, i.e. all the

finite set s ofreal number s $t\_i$ 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 anintegrable 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 byWladislav Orlicz andJulian Musielak in the paper Harv|Musielak|Orlicz|1959:Laurence 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" where introduced byLuigi Ambrosio andEnnio de Giorgi in the paper Harv|Ambrosio|De Giorgi|1988, dealing with free discontinuityvariational problem s: given anopen subset **$Omega$**of $scriptstylemathbb\{R\}^n$, the space**$SBV(Omega)$**is a propersubspace of**$BV(Omega)$**, since the weakgradient of each function belonging to it const exatcly of thesum of a $n$-dimension al support and a $n-1$-dimension al support measure and "no lower-dimensional terms", as seen in the following definition.**Definition**. Given alocally integrable function **$u$**, then $scriptstyle\; uin\; \{S!BV\}(Omega)$ if and only if**1.**There exist twoBorel function s $f$ and $g$ of domain**$Omega$**andcodomain $scriptstyle\; mathbb\{R\}^n$ 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::$int\_Omega\; umbox\{div\}\; phi\; dH^n\; =\; int\_Omega\; langle\; phi,\; f\; angle\; dH^n\; +int\_Omega\; langle\; phi,\; g\; angle\; dH^\{n-1\}.$

where $H^alpha$ is the $alpha$-

dimension alHausdorff measure .Details on the properties of "SBV" functions can be found in works cited in the bibliography section: particularly the paper Harv|De Giorgi|1992 contains a useful

bibliography .**"bv" sequences**As particular examples of

Banach spaces , harvtxt|Dunford|Schwartz|1958|loc=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"=("x"_{i}) 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 "bv"

_{0}, consisting of sequences "x" = ("x"_{i}) for which:$lim\_\{n\; oinfty\}\; x\_n\; =0.$The norm on "bv"_{0}is denoted:$|x|\_\{bv\_0\}\; =\; TV(x)\; =\; sum\_\{i=1\}^infty\; |x\_\{i+1\}-x\_i|.$With respect to this norm "bv"_{0}becomes a Banach space as well.**Examples**The function

:$f(x)\; =\; egin\{cases\}\; 0,\; mbox\{if\; \}x\; =0\; \backslash \; sin(1/x),\; mbox\{if\; \}\; x\; eq\; 0\; end\{cases\}$

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

While it is harder to see, the function

:$f(x)\; =\; egin\{cases\}\; 0,\; mbox\{if\; \}x\; =0\; \backslash \; x\; sin(1/x),\; mbox\{if\; \}\; x\; eq\; 0\; end\{cases\}$

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

At the same time, the function

:$f(x)\; =\; egin\{cases\}\; 0,\; mbox\{if\; \}x\; =0\; \backslash \; x^2\; sin(1/x),\; mbox\{if\; \}\; x\; eq\; 0\; end\{cases\}$

"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 aproper subset of**$BV(Omega)$**. In fact, for each**$u$**in**$W^\{1,1\}(Omega)$**it is possible to choose a measure $scriptstyle\; mu:=\; abla\; u\; mathcal\; L$ (where $scriptstylemathcal\; L$ is theLebesgue measure on**$Omega$**) such that the equality:$int\; umathrm\{div\}phi\; =\; -int\; phi,\; dmu\; =\; -int\; phi\; abla\; u\; qquad\; forall\; phiin\; C\_c^1$

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**$W^\{1,1\}$**.**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 ;

* $f$ hasone-sided limit s everywhere (limits from the left everywhere in $(a,b]$, and from the right everywhere in ["a","b") );

* thederivative $f\text{'}(x)$ existsalmost everywhere (i.e. except for a set ofmeasure zero ).

*Minimal surface s turn out very often to be graphs of "BV" functions: in this context, see reference Harv|Giusti|1984.**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 Harv|Hudjaev|Vol'pert|1986 details a very ample set of mathematical physics applications of "BV" functions. Also there is some modern application which deserves a brief description.

*The

Mumford-Shah functional : the segmentation problem for a two-dimensional image, i.e. the problem of faithful reproduction of contours and grey scales is equivalent to the minimization of suchfunctional .**See also***

Total variation

*Caccioppoli set

* "L"^{"p"}(Ω) space

*Lebesgue-Stieltjes integral ,Riemann-Stieltjes integral

*Radon measure

*Reduced derivative

*Helly's selection theorem

*Renato Caccioppoli ,Lamberto Cesari ,Ennio de Giorgi **References***Harvrefcol

Surname = Cesari

Given = Lamberto

Year = 1986

Chapter = L'opera diLeonida Tonelli e la sua influenza nel pensiero scientifico del secolo (the work ofLeonida Tonelli and his influence on scientific thinking in this century)

Editor = G. Montalenti et als.

Title = Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli)

Publisher =Accademia Nazionale dei Lincei , [*http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32847 Atti dei Convegni Lincei, Vol. 77*]

URL = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847

Place = Rome, 6-9 June 1985

Pages = 41-73 . Some recollections from one of the founders of the theory of "BV" functions of several variables (in Italian).

*Harvrefcol

Surname1 = Dunford

Given1 = Nelson

Surname2 = Schwartz

Given2 = Jacob T.

Title = Linear operators. Part I: General Theory

Publisher =Wiley-Interscience

Place =New York -London -Sydney

Year = 1958. Includes a discussion of the functional-analytic properties of spaces of functions of bounded variation.

*Harvrefcol

Surname = Giusti

Given = Enrico

Title = Minimal surfaces and functions of bounded variations

Publisher = [*http://www.birkhauser.com Birkhäuser Verlag*]

Place =Basel

Year = 1984 ISBN 0-8176-3153-4, particularly part I, chapter 1 "Functions of bounded variation and Caccioppoli sets".

*Harvrefcol

Surname1 = Hudjaev

Given1 = Sergei Ivanovich

Surname2 = Vol'pert

Given2 = Aizik Isaakovich

Title = Analysis in classes of discontinuous functions and equations of mathematical physics

Publisher = Martinus Nijhoff Publishers

Place =Dordrecht

Year=1986 ISBN 90-247-3109-7. The whole book is devoted to the theory of "BV" functions and their applications to problems inmathematical physics involvingdiscontinuous function s and geometric objects with non-smooth boundaries.

*Harvrefcol

Surname1 = Kannan

Given1 = Rangachary

Surname2 = King Krueger

Given2 = Carole

Title = Advanced analysis on the real line

Publisher =Springer Verlag

Place =Berlin -Heidelberg -New York

Year = 1996 ISBN 0-387-94642-X. Maybe the most complete book reference for the theory of "BV" functions in one variable: classical results and advanced results are collected in chapter 6 "Bounded variation" along with several exercises. The first author was a collaborator ofLamberto Cesari .

*citation

first=Andrej N.

last=Kolmogorov

first2=Sergej V.

last2=Fomin

title=Introductory Real Analysis

publisher=Dover Publications

place=New York

year=1969

id=ISBN 0486612260

*Harvrefcol

Surname1 = Màlek

Given1 = Josef

Surname2 = Nečas

Given2 = Jindřich

Surname3 = Rokyta

Given3 = Mirko

Surname4 = Růžička

Given4 = Michael

Title = Weak and measure-valued solutions to evolutionary PDEs

Publisher = Chapman & Hall/CRC Press

Place =London -Weinheim -New York -Tokyo -Melbourne -Madras

Year = 1996 ISBN 0-412-57750-X. One of the most complete monographs on the theory ofYoung measure s, strongly oriented to applications in continuum mechanics of fluids.

* Harvrefcol

Surname = Moreau

Given = Jean Jacques

Year= 1988

Chapter = Bounded variation in time.

Editor= Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G.

Title = Topics in nonsmooth mechanics.

Pages = 1-74

Publisher = [*http://www.birkhauser.com Birkhäuser Verlag*]

Place =Basel ISBN 3-7643-1907-0.

*Harvrefcol

Surname1 = Musielak

Given1 = Julian

Surname2 = Orlicz

Given2 = Wladislaw

Title = [*http://matwbn.icm.edu.pl/ksiazki/sm/sm18/sm1812.pdf On generalized variations (I)*]

Journal = [*http://matwbn.icm.edu.pl/spis.php?wyd=2&jez= Studia Mathematica*]

Volume = 18

Page = 13-41

Place =Warszawa -Wrocław

Year = 1959 . The first paper where weighted "BV" functions are studied in full generality.

*citation

first=Frigyes

last=Riesz

first2=Béla

last2=Szőkefalvi-Nagy

title=Functional Analysis

publisher=Dover Publications

place=New York

year=1990

id=ISBN 0-486-66289-6

*Harvrefcol

Surname = Vol'pert

Given = Aizik Isaakovich

Title = [*http://www.math.technion.ac.il/~volp/spaces_BV.pdf Spaces BV and quasi-linear equations*]

Journal = [*http://www.turpion.org/php/homes/pa.phtml?jrnid=sm Mathematics USSR-Sbornik*]

Volume = 2

Issue = 2

Page = 225-267

Year = 1967

Access-date =January 23 ,2007 . A seminal paper whereCaccioppoli set s and "BV" functions are deeply studied and applied to the theory ofpartial differential equation s.**Bibliography***Harvrefcol

Surname1 = Alberti

Given1 = Giovanni

Surname2 = Mantegazza

Given2 = Carlo

Title = A note on the theory of SBV functions

Journal = [*http://umi.dm.unibo.it/italiano/Editoria/BollettinoB.html Bollettino Unione Matematica Italiana, Sezione B*]

Volume = 7

Year = 1997

Page = 375-382. A paper containing a demonstration of the compactness of the set of SBV functions.

*Harvrefcol

Surname1 = Ambrosio

Given1 = Luigi

Surname2 = Dal Maso

Given2 = Giovanni

Title = [*http://links.jstor.org/sici?sici=0002-9939%28199003%29108%3A3%3C691%3AAGCRFD%3E2.0.CO%3B2-3&size=LARGE&origin=JSTOR-enlargePage A General Chain Rule for Distributional Derivatives*]

Journal = [*http://www.ams.org/journals/proc/ Proceedings of the American Mathematical Society*]

Volume = 108

Issue = 3

Year = 1990

Page = 691-702. DOI 10.2307/2047789. A paper containing a very generalchain rule formula for composition of BV functions.

*Harvrefcol

Surname1 = Ambrosio

Given1 = Luigi

Surname2 = De Giorgi

Given2 = Ennio

Title = Un nuovo tipo di funzionale del calcolo delle variazioni (A new kind of functional in thecalculus of variations )

Journal = Atti dell'Accademia Nazionale dei Lincei , [*http://www.lincei.it/pubblicazioni/rendicontiFMN/cliccami_eng.htm Rendiconti Lincei, Classe di Scienze Fisiche, Mathematiche, Naturali*]

Volume = 82

Year = 1988

Page = 199-210 (in Italian). The first paper about "SBV" functions and related variational problems.

*Harvrefcol

Surname1 = Conway

Given1 = Edward D

Surname2 = Smoller

Given2 = Joel A.

Title = Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables

Journal = [*http://www3.interscience.wiley.com/cgi-bin/jhome/29240 Communications on Pure and Applied Mathematics*]

Volume = 19

Year = 1966

Page = 95-105. An important paper where properties of "BV" functions were applied to "single"hyperbolic equation s of first order.

*Harvrefcol

Surname = De Giorgi

Given = Ennio

Year = 1992

Chapter = Problemi variazionali con discontinuità libere (Free-discontinuity variational problems)

Editor = E. Amaldi et als.

Title = Convegno internazionale in memoria diVito Volterra (International congress in memory ofVito Volterra )

Publisher =Accademia Nazionale dei Lincei , [*http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862 Atti dei Convegni Lincei, Vol. 92*]

Place = Roma, 8-11 October 1990

Pages = 133-150 . A survey paper on free-discontinuity variational problems including several details on the theory of "SBV" functions, their applications and a rich bibliography (in Italian), written byEnnio de Giorgi .*Tony F. Chan and Jackie (Jianhong) Shen (2005), [

*http://jackieneoshen.googlepages.com/ImagingNewEra.html "Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods"*] , SIAM Publisher, ISBN 089871589X (with in-depth coverage and extensive applications of Bounded Variations in modern image processing, as started by Rudin, Osher, and Fatemi).**External links****Theory*** Boris I. Golubov (and comments of

Anatolii Georgievich Vitushkin ) " [*http://eom.springer.de/V/v096110.htm Variation of a function*] ", Springer-Verlag Online Encyclopaedia of Mathematics.

*.

*Harvrefcol

Surname = Jordan

Given = Camille

Title = [*http://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227.chemindefer Sur la série de Fourier*]

Journal = Comptes rendus des Académie des sciences de Paris

Volume = 92

Place =Paris , janv.-juin

Year = 1881

Page = 228-230

Access-date=January 23 ,2007 (atGallica ). This is, according to Boris Golubov, the first paper on functions of bounded variation.

* Rowland, Todd and Weisstein, Eric W. " [*http://mathworld.wolfram.com/BoundedVariation.html Bounded Variation*] ". From MathWorld—A Wolfram Web Resource.**Other*** Luigi Ambrosio [

*http://cvgmt.sns.it/people/ambrosio/ home page*] at theScuola Normale Superiore ,Pisa . Academic home page (with preprints and publications of one of the contributors to the theory and applications of BV functions.

* [*http://cvgmt.sns.it/ Research Group in Calculus of Variations and Geometric Measure Theory*] ,Scuola Normale Superiore ,Pisa .

* [*http://www.math.technion.ac.il/people/volp/index.html Aizik Isaakovich Vol'pert*] atTechnion . Academic home page of one of the leading contributors to the theory of "BV" functions.----

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