Inmathematical analysis, a function ofbounded variation, also known asBV function, is areal-valuedfunction 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 thedirection of they-axis, neglecting the contribution of motion alongx-axis, traveled by apoint 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 ahypersurface in this case), but can be everyintersection of the graph itself with ahyperplane (in the case of functions of two variables, aplane) parallel to a fixedx-axis and to they-axis.

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

Another characterization states that the functions of bounded variation on a compact interval are exactly thosef which can be written as a differenceg − h, where bothg andh are boundedmonotone. In particular, a BV function may have discontinuities, but at most countably many.

In the case of several variables, a functionf defined on anopen subsetΩ of${\displaystyle {\mathbb{R}}^n}$ is said to have bounded variation if itsdistributional derivative is avector-valued finiteRadon measure.

One of the most important aspects of functions of bounded variation is that they form analgebra ofdiscontinuous functions whose first derivative existsalmost everywhere: due to this fact, they can and frequently are used to definegeneralized solutions of nonlinear problems involvingfunctionals andordinary andpartial differential equations inmathematics,physics andengineering.

We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:

Continuously differentiable ⊆Lipschitz continuous ⊆absolutely continuous ⊆continuous and bounded variation ⊆differentiablealmost everywhere

According to Boris Golubov, BV functions of a single variable were first introduced byCamille Jordan, in the paper (Jordan 1881) dealing with the convergence ofFourier series. "The properties of functions of bounded variation became widely known because they were discussed by Jordan in a note appended to the third volume of hisCourse d’analyse (1887).^[1]

The first successful step in the generalization of this concept to functions of several variables was due toLeonida Tonelli,^[2] who introduced a class ofcontinuous BV functions in 1926 (Cesari 1986, pp. 47–48), to extend hisdirect method for finding solutions to problems in thecalculus of variations in more than one variable. Ten years after, in (Cesari 1936),Lamberto Cesarichanged the continuity requirement in Tonelli's definitionto a less restrictiveintegrability 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 studyFourier series in several variables,geometric measure theory, calculus of variations, andmathematical physics.Renato Caccioppoli andEnnio De Giorgi used them to definemeasure ofnonsmoothboundaries ofsets (see the entry "Caccioppoli set" for further information).Olga Arsenievna Oleinik introduced her view of generalized solutions fornonlinear 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 afirst order partial differential equation in the paper (Oleinik 1959): few years later,Edward D. Conway andJoel A. Smoller applied BV-functions to the study of a singlenonlinear hyperbolic partial differential equation of first order in the paper (Conway & Smoller 1966), proving that the solution of theCauchy problem for such equations is a function of bounded variation, provided theinitial 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 thechain rule for BV functions and in the book (Hudjaev & Vol'pert 1985) he, jointly with his pupilSergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended byLuigi Ambrosio andGianni Dal Maso in the paper (Ambrosio & Dal Maso 1990).

Definition 1.1. Thetotal variation of areal-valued (or more generallycomplex-valued)functionf, defined on aninterval${\displaystyle [a,b]\subset \mathbb{R}}$ is the quantity

${\displaystyle V_a^b(f)=\underset{P\in \mathcal{P}}{sup}\sum _{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|.}$

where thesupremum is taken over the set$\mathcal{P}=\left\{P=\{x_0,\dots ,x_{n_P}\}\mid P\text{ is a partition of }[a,b]\text{ satisfying }{x}_i\le x_{i+1}\text{ for }0\le i\le n_P-1\right\}$ of allpartitions of the interval considered.

Iff isdifferentiable and its derivative is Riemann-integrable, its total variation is the vertical component of thearc-length of its graph, that is to say,

${\displaystyle V_a^b(f)={\int }_a^b|f^{\prime }(x)|\mathrm{d}x.}$

Definition 1.2. A real-valued function${\displaystyle f}$ on thereal line is said to be ofbounded variation (BV function) on a choseninterval${\displaystyle [a,b]\subset \mathbb{R}}$ if its total variation is finite,i.e.

It can be proved that a real function${\displaystyle f}$ is of bounded variation in${\displaystyle [a,b]}$ if and only if it can be written as the difference${\displaystyle f=f_1-f_2}$ of two non-decreasing functions${\displaystyle f_1}$ and${\displaystyle f_2}$ on${\displaystyle [a,b]}$: this result is known as theJordan decomposition of a function and it is related to theJordan decomposition of a measure.

Through theStieltjes integral, any function of bounded variation on a closed interval${\displaystyle [a,b]}$ defines abounded linear functional on${\displaystyle C([a,b])}$. In this special case,^[3] theRiesz–Markov–Kakutani representation theorem states that every bounded linear functional arises uniquely in this way. The normalized positive functionals orprobability measures correspond to positive non-decreasing lowersemicontinuous functions. This point of view has been important inspectral theory,^[4] in particular in its application toordinary differential equations.

Functions of bounded variation, BVfunctions, are functions whose distributionalderivative is afinite^[5]Radon measure. More precisely:

Definition 2.1. Let${\displaystyle \mathrm{\Omega }}$ be anopen subset of${\displaystyle {\mathbb{R}}^n}$. A function${\displaystyle u}$ belonging to${\displaystyle L^1(\mathrm{\Omega })}$ is said to be ofbounded variation (BV function), and written

${\displaystyle u\in \mathrm{BV}(\mathrm{\Omega })}$

if there exists afinitevectorRadon measure${\displaystyle Du\in \mathcal{M}(\mathrm{\Omega },{\mathbb{R}}^n)}$ such that the following equality holds

${\displaystyle {\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x=-{\int }_{\mathrm{\Omega }}⟨\mathit{\varphi },Du(x)⟩\mathrm{\forall }\mathit{\varphi }\in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$

that is,${\displaystyle u}$ defines alinear functional on the space${\displaystyle C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$ ofcontinuously differentiablevector functions${\displaystyle \mathit{\varphi }}$ ofcompact support contained in${\displaystyle \mathrm{\Omega }}$: the vectormeasure${\displaystyle Du}$ represents therefore thedistributional orweakgradient of${\displaystyle u}$.

BV can be defined equivalently in the following way.

Definition 2.2. Given a function${\displaystyle u}$ belonging to${\displaystyle L^1(\mathrm{\Omega })}$, thetotal variation of${\displaystyle u}$^[6] in${\displaystyle \mathrm{\Omega }}$ is defined as

${\displaystyle V(u,\mathrm{\Omega }):=sup\left\{{\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x:\mathit{\varphi }\in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n),\Vert \mathit{\varphi }{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le 1\right\}}$

where${\displaystyle \Vert {\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}}$ is theessential supremumnorm. Sometimes, especially in the theory ofCaccioppoli sets, the following notation is used

${\displaystyle {\int }_{\mathrm{\Omega }}|Du|=V(u,\mathrm{\Omega })}$

in order to emphasize that${\displaystyle V(u,\mathrm{\Omega })}$ is the total variation of thedistributional /weakgradient of${\displaystyle u}$. This notation reminds also that if${\displaystyle u}$ is of class${\displaystyle C^1}$ (i.e. acontinuous anddifferentiable function havingcontinuousderivatives) then itsvariation is exactly theintegral of theabsolute value of itsgradient.

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

The two definitions are equivalent since if then

${\displaystyle \left|{\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x\right|\le V(u,\mathrm{\Omega })\Vert \mathit{\varphi }{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\mathrm{\forall }\mathit{\varphi }\in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$

therefore${\displaystyle \mathit{\varphi }\mapsto {\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)dx}$ defines acontinuous linear functional on the space${\displaystyle C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$. Since${\displaystyle C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)\subset C^0(\mathrm{\Omega },{\mathbb{R}}^n)}$ as alinear subspace, thiscontinuous linear functional can be extendedcontinuously andlinearly to the whole${\displaystyle C^0(\mathrm{\Omega },{\mathbb{R}}^n)}$ by theHahn–Banach theorem. Hence the continuous linear functional defines aRadon measure by theRiesz–Markov–Kakutani representation theorem.

If thefunction space oflocally integrable functions, i.e.functions belonging to${\displaystyle L_{\text{loc}}^1(\mathrm{\Omega })}$, is considered in the preceding definitions1.2,2.1 and2.2 instead of the one ofglobally integrable functions, then the function space defined is that offunctions of locally bounded variation. Precisely, developing this idea fordefinition 2.2, alocal variation is defined as follows,

${\displaystyle V(u,U):=sup\left\{{\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x:\mathit{\varphi }\in C_c^1(U,{\mathbb{R}}^n),\Vert \mathit{\varphi }{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le 1\right\}}$

for everyset${\displaystyle U\in {\mathcal{O}}_c(\mathrm{\Omega })}$, having defined${\displaystyle {\mathcal{O}}_c(\mathrm{\Omega })}$ as the set of allprecompactopen subsets of${\displaystyle \mathrm{\Omega }}$ with respect to the standardtopology offinite-dimensionalvector spaces, and correspondingly the class of functions of locally bounded variation is defined as

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 referencesGiusti (1984) (partially),Hudjaev & Vol'pert (1985) (partially),Giaquinta, Modica & Souček (1998) and is the following one

• ${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ identifies thespace of functions of globally bounded variation
• ${\displaystyle {\mathrm{BV}}_{\text{loc}}(\mathrm{\Omega })}$ identifies thespace of functions of locally bounded variation

The second one, which is adopted in referencesVol'pert (1967) andMaz'ya (1985) (partially), is the following:

• ${\displaystyle \overline{\mathrm{BV}}(\mathrm{\Omega })}$ identifies thespace of functions of globally bounded variation
• ${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ identifies thespace of functions of locally bounded variation

Only the properties common tofunctions of one variable and tofunctions of several variables will be considered in the following, andproofs 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: 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.

In the case of one variable, the assertion is clear: for each point${\displaystyle x_0}$ in theinterval${\displaystyle [a,b]\subset \mathbb{R}}$ of definition of the function${\displaystyle u}$, either one of the following two assertions is true

${\displaystyle \underset{x\to x_{0^{-}}}{lim}u(x)=\underset{x\to x_{0^{+}}}{lim}u(x)}$

${\displaystyle \underset{x\to x_{0^{-}}}{lim}u(x)\ne \underset{x\to x_{0^{+}}}{lim}u(x)}$

while bothlimits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is acontinuum ofdirections along which it is possible to approach a given point${\displaystyle x_0}$ belonging to the domain${\displaystyle \mathrm{\Omega }}$⊂${\displaystyle {\mathbb{R}}^n}$. It is necessary to make precise a suitable concept oflimit: choosing aunit vector${\displaystyle \hat{\mathit{a}}\in {\mathbb{R}}^n}$ it is possible to divide${\displaystyle \mathrm{\Omega }}$ in two sets

${\displaystyle {\mathrm{\Omega }}_{(\hat{\mathit{a}},{\mathit{x}}_0)}=\mathrm{\Omega }\cap \{\mathit{x}\in {\mathbb{R}}^n|⟨\mathit{x}-{\mathit{x}}_0,\hat{\mathit{a}}⟩>0\}{\mathrm{\Omega }}_{(-\hat{\mathit{a}},{\mathit{x}}_0)}=\mathrm{\Omega }\cap \{\mathit{x}\in {\mathbb{R}}^n|⟨\mathit{x}-{\mathit{x}}_0,-\hat{\mathit{a}}⟩>0\}}$

Then for each point${\displaystyle x_0}$ belonging to the domain${\displaystyle \mathrm{\Omega }\in {\mathbb{R}}^n}$ of the BV function${\displaystyle u}$, only one of the following two assertions is true

${\displaystyle \underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})=\underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(-\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})}$

${\displaystyle \underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})\ne \underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(-\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})}$

or${\displaystyle x_0}$ belongs to asubset of${\displaystyle \mathrm{\Omega }}$ having zero${\displaystyle n-1}$-dimensionalHausdorff measure. The quantities

${\displaystyle \underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})=u_{\hat{\mathit{a}}}({\mathit{x}}_0)\underset{\stackrel{\mathit{x}\to {\mathit{x}}_0}{\mathit{x}\in {\mathrm{\Omega }}_{(-\hat{\mathit{a}},{\mathit{x}}_0)}}}{lim}u(\mathit{x})=u_{-\hat{\mathit{a}}}({\mathit{x}}_0)}$

are calledapproximate limits of the BV function${\displaystyle u}$ at the point${\displaystyle x_0}$.

Thefunctional${\displaystyle V(\cdot ,\mathrm{\Omega }):\mathrm{BV}(\mathrm{\Omega })\to {\mathbb{R}}^{+}}$ islower semi-continuous:to see this, choose aCauchy sequence of BV-functions${\displaystyle \{u_n{\}}_{n\in \mathbb{N}}}$ converging to${\displaystyle u\in L_{\text{loc}}^1(\mathrm{\Omega })}$. Then, since all the functions of the sequence and their limit function areintegrable and by the definition oflower limit

Now considering thesupremum on the set of functions${\displaystyle \mathit{\varphi }\in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$ such that${\displaystyle \Vert \mathit{\varphi }{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le 1}$ then the following inequality holds true

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}V(u_n,\mathrm{\Omega })\ge V(u,\mathrm{\Omega })}$

which is exactly the definition oflower semicontinuity.

By definition${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ is asubset of${\displaystyle L^1(\mathrm{\Omega })}$, whilelinearity follows from the linearity properties of the definingintegral i.e.

for all${\displaystyle \varphi \in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$ therefore${\displaystyle u+v\in \mathrm{BV}(\mathrm{\Omega })}$for all${\displaystyle u,v\in \mathrm{BV}(\mathrm{\Omega })}$, and

${\displaystyle {\int }_{\mathrm{\Omega }}c\cdot u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x=c{\int }_{\mathrm{\Omega }}u(x)\mathrm{div}\mathit{\varphi }(x)\mathrm{d}x=-c{\int }_{\mathrm{\Omega }}⟨\mathit{\varphi }(x),Du(x)⟩}$

for all${\displaystyle c\in \mathbb{R}}$, therefore${\displaystyle cu\in \mathrm{BV}(\mathrm{\Omega })}$ for all${\displaystyle u\in \mathrm{BV}(\mathrm{\Omega })}$, and all${\displaystyle c\in \mathbb{R}}$. The provedvector space properties imply that${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ is avector subspace of${\displaystyle L^1(\mathrm{\Omega })}$. Consider now the function${\displaystyle \Vert {\Vert }_{\mathrm{BV}}:\mathrm{BV}(\mathrm{\Omega })\to {\mathbb{R}}^{+}}$ defined as

${\displaystyle \Vert u{\Vert }_{\mathrm{BV}}:=\Vert u{\Vert }_{L^1}+V(u,\mathrm{\Omega })}$

where${\displaystyle \Vert {\Vert }_{L^1}}$ is the usual${\displaystyle L^1(\mathrm{\Omega })}$ norm: it is easy to prove that this is anorm on${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$. To see that${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ iscomplete respect to it, i.e. it is aBanach space, consider aCauchy sequence${\displaystyle \{u_n{\}}_{n\in \mathbb{N}}}$ in${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$. By definition it is also aCauchy sequence in${\displaystyle L^1(\mathrm{\Omega })}$ and therefore has alimit${\displaystyle u}$ in${\displaystyle L^1(\mathrm{\Omega })}$: since${\displaystyle u_n}$ is bounded in${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$ for each${\displaystyle n}$, then bylower semicontinuity of the variation${\displaystyle V(\cdot ,\mathrm{\Omega })}$, therefore${\displaystyle u}$ is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number${\displaystyle \epsilon }$

From this we deduce that${\displaystyle V(\cdot ,\mathrm{\Omega })}$ is continuous because it's a norm.

To see this, it is sufficient to consider the following example belonging to the space${\displaystyle \mathrm{BV}([0,1])}$:^[7] for each 0 < α < 1 define

as thecharacteristic function of theleft-closed interval${\displaystyle [\alpha ,1]}$. Then, choosing${\displaystyle \alpha ,\beta \in [0,1]}$ such that${\displaystyle \alpha \ne \beta }$ the following relation holds true:

${\displaystyle \Vert {\chi }_{\alpha }-{\chi }_{\beta }{\Vert }_{\mathrm{BV}}=2}$

Now, in order to prove that everydense subset of${\displaystyle \mathrm{BV}(]0,1[)}$ cannot becountable, it is sufficient to see that for every${\displaystyle \alpha \in [0,1]}$ it is possible to construct theballs

${\displaystyle B_{\alpha }=\left\{\psi \in \mathrm{BV}([0,1]);\Vert {\chi }_{\alpha }-\psi {\Vert }_{\mathrm{BV}}\le 1\right\}}$

Obviously those balls arepairwise disjoint, and also are anindexed family ofsets whoseindex set is${\displaystyle [0,1]}$. This implies that this family has thecardinality of the continuum: now, since every dense subset of${\displaystyle \mathrm{BV}([0,1])}$ 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.^[8] This example can be obviously extended to higher dimensions, and since it involves onlylocal properties, it implies that the same property is true also for${\displaystyle {\mathrm{BV}}_{loc}}$.

Chain rules fornonsmooth functions are very important inmathematics andmathematical physics since there are several importantphysical models whose behaviors are described byfunctions orfunctionals with a very limited degree ofsmoothness. The following chain rule is proved in the paper (Vol'pert 1967, p. 248). Note allpartial derivatives must be interpreted in a generalized sense, i.e., asgeneralized derivatives.

Theorem. Let${\displaystyle f:{\mathbb{R}}^p\to \mathbb{R}}$ be a function of class${\displaystyle C^1}$ (i.e. acontinuous anddifferentiable function havingcontinuousderivatives) and let${\displaystyle \mathit{u}(\mathit{x})=(u_1(\mathit{x}),\dots ,u_p(\mathit{x}))}$ be a function in${\displaystyle {\mathrm{BV}}_{loc}(\mathrm{\Omega })}$ with${\displaystyle \mathrm{\Omega }}$ being anopen subset of${\displaystyle {\mathbb{R}}^n}$.Then${\displaystyle f\circ \mathit{u}(\mathit{x})=f(\mathit{u}(\mathit{x}))\in {\mathrm{BV}}_{loc}(\mathrm{\Omega })}$ and

${\displaystyle \frac{\mathrm{\partial }f(\mathit{u}(\mathit{x}))}{\mathrm{\partial }{x}_i}=\sum _{k=1}^p\frac{\mathrm{\partial }\overline{f}(\mathit{u}(\mathit{x}))}{\mathrm{\partial }{u}_k}\frac{\mathrm{\partial }{u}_k(\mathit{x})}{\mathrm{\partial }{x}_i}\mathrm{\forall }i=1,\dots ,n}$

where${\displaystyle \overline{f}(\mathit{u}(\mathit{x}))}$ is the mean value of the function at the point${\displaystyle x\in \mathrm{\Omega }}$, defined as

${\displaystyle \overline{f}(\mathit{u}(\mathit{x}))={\int }_0^{1}f\left({\mathit{u}}_{\hat{\mathit{a}}}(\mathit{x})t+{\mathit{u}}_{-\hat{\mathit{a}}}(\mathit{x})(1-t)\right)dt}$

A more generalchain ruleformula forLipschitz continuous functions${\displaystyle f:{\mathbb{R}}^p\to {\mathbb{R}}^s}$ has been found byLuigi Ambrosio andGianni Dal Maso and is published in the paper (Ambrosio & Dal Maso 1990). However, even this formula has very important direct consequences: we use${\displaystyle (u(\mathit{x}),v(\mathit{x}))}$ in place of${\displaystyle \mathit{u}(\mathit{x})}$, where${\displaystyle v(\mathit{x})}$ is also a${\displaystyle B{V}_{loc}}$ function. We have to assume also that${\displaystyle \overline{u}(\mathit{x})}$ is locally integrable with respect to the measure${\displaystyle \frac{\mathrm{\partial }v(\mathit{x})}{\mathrm{\partial }{x}_i}}$ for each${\displaystyle i}$, and that${\displaystyle \overline{v}(\mathit{x})}$ is locally integrable with respect to the measure${\displaystyle \frac{\mathrm{\partial }u(\mathit{x})}{\mathrm{\partial }{x}_i}}$ for each${\displaystyle i}$. Then choosing${\displaystyle f((u,v))=uv}$, the preceding formula gives theLeibniz rule for 'BV' functions

${\displaystyle \frac{\mathrm{\partial }v(\mathit{x})u(\mathit{x})}{\mathrm{\partial }{x}_i}=\overline{u}(\mathit{x})\frac{\mathrm{\partial }v(\mathit{x})}{\mathrm{\partial }{x}_i}+\overline{v}(\mathit{x})\frac{\mathrm{\partial }u(\mathit{x})}{\mathrm{\partial }{x}_i}}$

It is possible to generalize the above notion oftotal variation so that different variations are weighted differently. More precisely, let${\displaystyle \phi :[0,+\mathrm{\infty })⟶[0,+\mathrm{\infty })}$ be any increasing function such that${\displaystyle \phi (0)=\phi (0+)=\underset{x\to 0_{+}}{lim}\phi (x)=0}$ (theweight function) and let${\displaystyle f:[0,T]⟶X}$ be a function from theinterval${\displaystyle [0,T]}$${\displaystyle \subset \mathbb{R}}$ taking values in anormed vector space${\displaystyle X}$. Then the${\displaystyle \mathit{\phi }}$-variation of${\displaystyle f}$ over${\displaystyle [0,T]}$ is defined as

${\displaystyle \underset{[0,T]}{\phi \text{-}\mathrm{Var}}(f):=sup\sum _{j=0}^k\phi \left(|f(t_{j+1})-f(t_j){|}_X\right),}$

where, as usual, the supremum is taken over all finitepartitions of the interval${\displaystyle [0,T]}$, i.e. all thefinite sets ofreal numbers${\displaystyle t_i}$ such that

The original notion ofvariation considered above is the special case of${\displaystyle \phi }$-variation for which the weight function is theidentity function: therefore anintegrable function${\displaystyle f}$ is said to be aweighted BV function (of weight${\displaystyle \phi }$) if and only if its${\displaystyle \phi }$-variation is finite.

The space${\displaystyle {\mathrm{BV}}_{\phi }([0,T];X)}$ is atopological vector space with respect to thenorm

${\displaystyle \Vert f{\Vert }_{{\mathrm{BV}}_{\phi }}:=\Vert f{\Vert }_{\mathrm{\infty }}+\underset{[0,T]}{\phi \text{-}\mathrm{Var}}(f),}$

where${\displaystyle \Vert f{\Vert }_{\mathrm{\infty }}}$ denotes the usualsupremum norm of${\displaystyle f}$. Weighted BV functions were introduced and studied in full generality byWładysław Orlicz andJulian Musielak in the paperMusielak & Orlicz 1959:Laurence Chisholm Young studied earlier the case${\displaystyle \phi (x)=x^p}$ where${\displaystyle p}$ is a positive integer.

SBV functionsi.e.Special functions of Bounded Variation were introduced byLuigi Ambrosio andEnnio De Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuityvariational problems: given anopen subset${\displaystyle \mathrm{\Omega }}$ of${\displaystyle {\mathbb{R}}^n}$, the space${\displaystyle \mathrm{SBV}(\mathrm{\Omega })}$ is a properlinear subspace of${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$, since theweakgradient of each function belonging to it consists precisely of thesum of an${\displaystyle n}$-dimensionalsupport and an${\displaystyle n-1}$-dimensionalsupportmeasure andno intermediate-dimensional terms, as seen in the following definition.

Definition. Given alocally integrable function${\displaystyle u}$, then${\displaystyle u\in \mathrm{SBV}(\mathrm{\Omega })}$ if and only if

1. There exist twoBorel functions${\displaystyle f}$ and${\displaystyle g}$ ofdomain${\displaystyle \mathrm{\Omega }}$ andcodomain${\displaystyle {\mathbb{R}}^n}$ such that

2. For all ofcontinuously differentiablevector functions${\displaystyle \varphi }$ ofcompact support contained in${\displaystyle \mathrm{\Omega }}$,i.e. for all${\displaystyle \varphi \in C_c^1(\mathrm{\Omega },{\mathbb{R}}^n)}$ the following formula is true:

${\displaystyle {\int }_{\mathrm{\Omega }}u\mathrm{div}\varphi d{H}^n={\int }_{\mathrm{\Omega }}⟨\varphi ,f⟩d{H}^n+{\int }_{\mathrm{\Omega }}⟨\varphi ,g⟩d{H}^{n-1}.}$

where${\displaystyle H^{\alpha }}$ is the${\displaystyle \alpha }$-dimensionalHausdorff measure.

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

As particular examples ofBanach spaces,Dunford & Schwartz (1958, Chapter IV) consider spaces ofsequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of asequencex = (x_i) of real or complex numbers is defined by

${\displaystyle \mathrm{TV}(x)=\sum _{i=1}^{\mathrm{\infty }}|x_{i+1}-x_i|.}$

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

${\displaystyle \Vert x{\Vert }_{\mathrm{BV}}=|x_1|+\mathrm{TV}(x)=|x_1|+\sum _{i=1}^{\mathrm{\infty }}|x_{i+1}-x_i|.}$

With this norm, the space BV is a Banach space which is isomorphic to${\displaystyle {\ell }_1}$.

The total variation itself defines a norm on a certain subspace of BV, denoted by BV₀, consisting of sequencesx = (x_i) for which

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{x}_n=0.}$

The norm on BV₀ is denoted

${\displaystyle \Vert x{\Vert }_{{\mathrm{BV}}_0}=\mathrm{TV}(x)=\sum _{i=1}^{\mathrm{\infty }}|x_{i+1}-x_i|.}$

With respect to this norm BV₀ becomes a Banach space as well, which is isomorphicand isometric to${\displaystyle {\ell }_1}$ (although not in the natural way).

Asigned (orcomplex)measure${\displaystyle \mu }$ on ameasurable space${\displaystyle (X,\mathrm{\Sigma })}$ is said to be of bounded variation if itstotal variation${\displaystyle \Vert \mu \Vert =|\mu |(X)}$ is bounded: seeHalmos (1950, p. 123),Kolmogorov & Fomin (1969, p. 346) or the entry "Total variation" for further details.

As mentioned in the introduction, two large class of examples of BV functions are monotone functions, and absolutely continuous functions. For a negative example: the function

isnot of bounded variation on the interval${\displaystyle [0,2/\pi ]}$

While it is harder to see, the continuous function

isnot of bounded variation on the interval${\displaystyle [0,2/\pi ]}$ either.

At the same time, the function

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

Every monotone, bounded function is of bounded variation. For such a function${\displaystyle f}$ on the interval${\displaystyle [a,b]}$ and any partition${\displaystyle P=\{x_0,\dots ,x_{n_P}\}}$ of this interval, it can be seen that

${\displaystyle \sum _{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|=|f(b)-f(a)|}$

from the fact that the sum on the left istelescoping. From this, it follows that for such${\displaystyle f}$,

${\displaystyle V_a^b(f)=|f(b)-f(a)|.}$

In particular, the monotoneCantor function is a well-known example of a function of bounded variation that is notabsolutely continuous.^[9]

TheSobolev space${\displaystyle W^{1,1}(\mathrm{\Omega })}$ is aproper subset of${\displaystyle \mathrm{BV}(\mathrm{\Omega })}$. In fact, for each${\displaystyle u}$ in${\displaystyle W^{1,1}(\mathrm{\Omega })}$ it is possible to choose ameasure${\displaystyle \mu :=\mathrm{\nabla }u\mathcal{L}}$ (where${\displaystyle \mathcal{L}}$ is theLebesgue measure on${\displaystyle \mathrm{\Omega }}$) such that the equality

${\displaystyle \int u\mathrm{div}\varphi =-\int \varphi d\mu =-\int \varphi \mathrm{\nabla }u\mathrm{\forall }\varphi \in C_c^1}$

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

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

• ${\displaystyle f}$ iscontinuous except at most on acountable set;
• ${\displaystyle f}$ hasone-sided limits everywhere (limits from the left everywhere in${\displaystyle (a,b]}$, and from the right everywhere in${\displaystyle [a,b)}$ ;
• thederivative${\displaystyle f^{\prime }(x)}$ existsalmost everywhere (i.e. except for a set ofmeasure zero).

Forrealfunctions of several real variables

• thecharacteristic function of aCaccioppoli set is a BV function: BV functions lie at the basis of the modern theory of perimeters.
• Minimal surfaces aregraphs of BV functions: in this context, see reference (Giusti 1984).

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.

• TheMumford–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 theminimization of suchfunctional.
• Total variation denoising

• Golubov, Boris I.;Vitushkin, Anatolii G. (2001) [1994],"Variation of a function",Encyclopedia of Mathematics,EMS Press
• "BV function".PlanetMath..
• Rowland, Todd & Weisstein, Eric W."Bounded Variation".MathWorld.
• Function of bounded variation atEncyclopedia of Mathematics

• Luigi Ambrosiohome page at theScuola Normale Superiore di Pisa. Academic home page (with preprints and publications) of one of the contributors to the theory and applications of BV functions.
• Research Group in Calculus of Variations and Geometric Measure Theory,Scuola Normale Superiore di Pisa.

This article incorporates material from BV function onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Real analysis
• Calculus of variations
• Measure theory

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