Bounded variation

for every set ${\displaystyle U\in {\mathcal {O}}_{c}(\Omega )}$, having defined ${\displaystyle {\mathcal {O}}_{c}(\Omega )}$ as the set of all

${\displaystyle \Omega }$

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

• ${\displaystyle BV(\Omega )}$ identifies the space of functions of globally bounded variation
• ${\displaystyle BV_{\text{loc}}(\Omega )}$ 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:

• ${\displaystyle {\overline {BV}}(\Omega )}$ identifies the space of functions of globally bounded variation
• ${\displaystyle BV(\Omega )}$ 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 or removable discontinuities

In the case of one variable, the assertion is clear: for each point ${\displaystyle x_{0}}$ in the interval ${\displaystyle [a,b]\subset \mathbb {R} }$ of definition of the function ${\displaystyle u}$, either one of the following two assertions is true

${\displaystyle \lim _{x\rightarrow x_{0^{-}}}\!\!\!u(x)=\!\!\!\lim _{x\rightarrow x_{0^{+}}}\!\!\!u(x)}$

${\displaystyle \lim _{x\rightarrow x_{0^{-}}}\!\!\!u(x)\neq \!\!\!\lim _{x\rightarrow x_{0^{+}}}\!\!\!u(x)}$

while both

${\displaystyle x_{0}}$

${\displaystyle \Omega }$

${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \scriptstyle {\boldsymbol {\hat {a}}}\in \mathbb {R} ^{n}}$

${\displaystyle \Omega }$

${\displaystyle \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}=\Omega \cap \{{\boldsymbol {x}}\in \mathbb {R} ^{n}|\langle {\boldsymbol {x}}-{\boldsymbol {x}}_{0},{\boldsymbol {\hat {a}}}\rangle >0\}\qquad \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}=\Omega \cap \{{\boldsymbol {x}}\in \mathbb {R} ^{n}|\langle {\boldsymbol {x}}-{\boldsymbol {x}}_{0},-{\boldsymbol {\hat {a}}}\rangle >0\}}$

Then for each point ${\displaystyle x_{0}}$ belonging to the domain ${\displaystyle \scriptstyle \Omega \in \mathbb {R} ^{n}}$ of the BV function ${\displaystyle u}$, only one of the following two assertions is true

${\displaystyle \lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})=\!\!\!\!\!\!\!\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})}$

${\displaystyle \lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})\neq \!\!\!\!\!\!\!\lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})}$

or ${\displaystyle x_{0}}$ belongs to a subset of ${\displaystyle \Omega }$ having zero ${\displaystyle n-1}$-dimensional Hausdorff measure. The quantities

${\displaystyle \lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{({\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!u({\boldsymbol {x}})=u_{\boldsymbol {\hat {a}}}({\boldsymbol {x}}_{0})\qquad \lim _{\overset {{\boldsymbol {x}}\rightarrow {\boldsymbol {x}}_{0}}{{\boldsymbol {x}}\in \Omega _{(-{\boldsymbol {\hat {a}}},{\boldsymbol {x}}_{0})}}}\!\!\!\!\!\!\!u({\boldsymbol {x}})=u_{-{\boldsymbol {\hat {a}}}}({\boldsymbol {x}}_{0})}$

are called approximate limits of the BV function ${\displaystyle u}$ at the point ${\displaystyle x_{0}}$.

V(·, Ω) is lower semi-continuous on L¹(Ω)

The functional ${\displaystyle \scriptstyle V(\cdot ,\Omega ):BV(\Omega )\rightarrow \mathbb {R} ^{+}}$ is lower semi-continuous: to see this, choose a Cauchy sequence of BV-functions ${\displaystyle \scriptstyle \{u_{n}\}_{n\in \mathbb {N} }}$ converging to ${\displaystyle \scriptstyle u\in L_{\text{loc}}^{1}(\Omega )}$. Then, since all the functions of the sequence and their limit function are

${\displaystyle \liminf _{n\rightarrow \infty }V(u_{n},\Omega )\geq \liminf _{n\rightarrow \infty }\int _{\Omega }u_{n}(x)\operatorname {div} \,{\boldsymbol {\phi }}\,\mathrm {d} x\geq \int _{\Omega }\lim _{n\rightarrow \infty }u_{n}(x)\operatorname {div} \,{\boldsymbol {\phi }}\,\mathrm {d} x=\int _{\Omega }u(x)\operatorname {div} {\boldsymbol {\phi }}\,\mathrm {d} x\qquad \forall {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\quad \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1}$

Now considering the

${\displaystyle \scriptstyle {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$

${\displaystyle \scriptstyle \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1}$

${\displaystyle \liminf _{n\rightarrow \infty }V(u_{n},\Omega )\geq V(u,\Omega )}$

which is exactly the definition of

BV(Ω) is a Banach space

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

${\displaystyle {\begin{aligned}\int _{\Omega }[u(x)+v(x)]\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x&=\int _{\Omega }u(x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x+\int _{\Omega }v(x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=\\&=-\int _{\Omega }\langle {\boldsymbol {\phi }}(x),Du(x)\rangle -\int _{\Omega }\langle {\boldsymbol {\phi }}(x),Dv(x)\rangle =-\int _{\Omega }\langle {\boldsymbol {\phi }}(x),[Du(x)+Dv(x)]\rangle \end{aligned}}}$

for all ${\displaystyle \scriptstyle \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$ therefore ${\displaystyle \scriptstyle u+v\in BV(\Omega )}$for all ${\displaystyle \scriptstyle u,v\in BV(\Omega )}$, and

${\displaystyle \int _{\Omega }c\cdot u(x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=c\int _{\Omega }u(x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x=-c\int _{\Omega }\langle {\boldsymbol {\phi }}(x),Du(x)\rangle }$

for all ${\displaystyle c\in \mathbb {R} }$, therefore ${\displaystyle cu\in BV(\Omega )}$ for all ${\displaystyle u\in BV(\Omega )}$, and all ${\displaystyle c\in \mathbb {R} }$. The proved vector space properties imply that ${\displaystyle BV(\Omega )}$ is a

${\displaystyle L^{1}(\Omega )}$

${\displaystyle \scriptstyle \|\;\|_{BV}:BV(\Omega )\rightarrow \mathbb {R} ^{+}}$

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

where ${\displaystyle \scriptstyle \|\;\|_{L^{1}}}$ is the usual ${\displaystyle L^{1}(\Omega )}$ norm: it is easy to prove that this is a norm on ${\displaystyle BV(\Omega )}$. To see that ${\displaystyle BV(\Omega )}$ is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence ${\displaystyle \scriptstyle \{u_{n}\}_{n\in \mathbb {N} }}$ in ${\displaystyle BV(\Omega )}$. By definition it is also a Cauchy sequence in ${\displaystyle L^{1}(\Omega )}$ and therefore has a limit ${\displaystyle u}$ in ${\displaystyle L^{1}(\Omega )}$: since ${\displaystyle u_{n}}$ is bounded in ${\displaystyle BV(\Omega )}$ for each ${\displaystyle n}$, then ${\displaystyle \scriptstyle \Vert u\Vert _{BV}<+\infty }$ by

${\displaystyle \scriptstyle V(\cdot ,\Omega )}$

${\displaystyle u}$

${\displaystyle \scriptstyle \varepsilon }$

${\displaystyle \Vert u_{j}-u_{k}\Vert _{BV}<\varepsilon \quad \forall j,k\geq N\in \mathbb {N} \quad \Rightarrow \quad V(u_{k}-u,\Omega )\leq \liminf _{j\rightarrow +\infty }V(u_{k}-u_{j},\Omega )\leq \varepsilon }$

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

BV(Ω) is not separable

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

${\displaystyle \chi _{\alpha }=\chi _{[\alpha ,1]}={\begin{cases}0&{\mbox{if }}x\notin \;[\alpha ,1]\\1&{\mbox{if }}x\in [\alpha ,1]\end{cases}}}$

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

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

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

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

Obviously those balls are pairwise disjoint, and also are an indexed family of sets whose index set is ${\displaystyle [0,1]}$. This implies that this family has the cardinality of the continuum: now, since every dense subset of ${\displaystyle BV([0,1])}$ 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 also for ${\displaystyle BV_{loc}}$.

Chain rule for BV functions

Theorem. Let ${\displaystyle \scriptstyle f:\mathbb {R} ^{p}\rightarrow \mathbb {R} }$ be a function of class ${\displaystyle C^{1}}$ (i.e. a continuous and differentiable function having continuous derivatives) and let ${\displaystyle \scriptstyle {\boldsymbol {u}}({\boldsymbol {x}})=(u_{1}({\boldsymbol {x}}),\ldots ,u_{p}({\boldsymbol {x}}))}$ be a function in ${\displaystyle BV(\Omega )}$ with ${\displaystyle \Omega }$ being an

${\displaystyle \scriptstyle \mathbb {R} ^{n}}$

${\displaystyle \scriptstyle f\circ {\boldsymbol {u}}({\boldsymbol {x}})=f({\boldsymbol {u}}({\boldsymbol {x}}))\in BV(\Omega )}$

${\displaystyle {\frac {\partial f({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial x_{i}}}=\sum _{k=1}^{p}{\frac {\partial {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}{\partial u_{k}}}{\frac {\partial {u_{k}({\boldsymbol {x}})}}{\partial x_{i}}}\qquad \forall i=1,\ldots ,n}$

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

${\displaystyle {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))=\int _{0}^{1}f\left({\boldsymbol {u}}_{\boldsymbol {\hat {a}}}({\boldsymbol {x}})t+{\boldsymbol {u}}_{-{\boldsymbol {\hat {a}}}}({\boldsymbol {x}})(1-t)\right)\,dt}$

A more general chain rule formula for Lipschitz continuous functions ${\displaystyle \scriptstyle f:\mathbb {R} ^{p}\rightarrow \mathbb {R} ^{s}}$ 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: using ${\displaystyle (u({\boldsymbol {x}}),v({\boldsymbol {x}}))}$ in place of ${\displaystyle {\boldsymbol {u}}({\boldsymbol {x}})}$, where ${\displaystyle v({\boldsymbol {x}})}$ is also a ${\displaystyle BV}$ function and choosing ${\displaystyle f((u,v))=uv}$, the preceding formula gives the Leibniz rule for ${\displaystyle BV}$ functions

${\displaystyle {\frac {\partial v({\boldsymbol {x}})u({\boldsymbol {x}})}{\partial x_{i}}}={{\bar {u}}({\boldsymbol {x}})}{\frac {\partial v({\boldsymbol {x}})}{\partial x_{i}}}+{{\bar {v}}({\boldsymbol {x}})}{\frac {\partial u({\boldsymbol {x}})}{\partial x_{i}}}}$

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

BV(Ω) is a Banach algebra

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

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

therefore the ordinary

${\displaystyle BV(\Omega )}$

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 ${\displaystyle \scriptstyle \varphi :[0,+\infty )\longrightarrow [0,+\infty )}$ be any increasing function such that ${\displaystyle \scriptstyle \varphi (0)=\varphi (0+)=\lim _{x\rightarrow 0_{+}}\varphi (x)=0}$ (the weight function) and let ${\displaystyle \scriptstyle f:[0,T]\longrightarrow X}$ be a function from the interval ${\displaystyle [0,T]}$${\displaystyle \subset \mathbb {R} }$ taking values in a normed vector space ${\displaystyle X}$. Then the ${\displaystyle \scriptstyle {\boldsymbol {\varphi }}}$-variation of ${\displaystyle f}$ over ${\displaystyle [0,T]}$ is defined as

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

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

${\displaystyle 0=t_{0}<t_{1}<\cdots <t_{k}=T.}$

The original notion of variation considered above is the special case of ${\displaystyle \scriptstyle \varphi }$-variation for which the weight function is the

${\displaystyle f}$

${\displaystyle \scriptstyle \varphi }$

${\displaystyle \scriptstyle \varphi }$

${\displaystyle f\in BV_{\varphi }([0,T];X)\iff \mathop {\varphi {\text{-}}\operatorname {Var} } _{[0,T]}(f)<+\infty }$

The space ${\displaystyle \scriptstyle BV_{\varphi }([0,T];X)}$ is a topological vector space with respect to the norm

${\displaystyle \|f\|_{BV_{\varphi }}:=\|f\|_{\infty }+\mathop {\varphi {\text{-}}\operatorname {Var} } _{[0,T]}(f),}$

where ${\displaystyle \scriptstyle \|f\|_{\infty }}$ denotes the usual

${\displaystyle f}$

${\displaystyle \scriptstyle \varphi (x)=x^{p}}$

${\displaystyle p}$

SBV functions

SBV functions i.e. Special functions of Bounded Variation were introduced by

${\displaystyle \Omega }$

${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle SBV(\Omega )}$

${\displaystyle BV(\Omega )}$

${\displaystyle n}$

${\displaystyle n-1}$

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

1. There exist two

${\displaystyle f}$

${\displaystyle g}$

${\displaystyle \Omega }$

${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \int _{\Omega }\vert f\vert \,dH^{n}+\int _{\Omega }\vert g\vert \,dH^{n-1}<+\infty .}$

2. For all of

${\displaystyle \scriptstyle \phi }$ of

${\displaystyle \Omega }$

${\displaystyle \scriptstyle \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$

${\displaystyle \int _{\Omega }u\operatorname {div} \phi \,dH^{n}=\int _{\Omega }\langle \phi ,f\rangle \,dH^{n}+\int _{\Omega }\langle \phi ,g\rangle \,dH^{n-1}.}$

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

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

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

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

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 sequences x = (x_i) for which

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

The norm on bv₀ is denoted

${\displaystyle \|x\|_{bv_{0}}=TV(x)=\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|.}$

With respect to this norm bv₀ becomes a Banach space as well, which is isomorphic and isometric to ${\displaystyle \ell _{1}}$ (although not in the natural way).

Measures of bounded variation

A signed (or complex) measure ${\displaystyle \mu }$ on a

${\displaystyle (X,\Sigma )}$

${\displaystyle \scriptstyle \Vert \mu \Vert =|\mu |(X)}$

Examples

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

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

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

While it is harder to see, the continuous function

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

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

At the same time, the function

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

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

Cantor function is a well-known example of a function of bounded variation that is not absolutely continuous.^[8]

The Sobolev space ${\displaystyle W^{1,1}(\Omega )}$ is a

${\displaystyle BV(\Omega )}$

${\displaystyle u}$

${\displaystyle W^{1,1}(\Omega )}$

${\displaystyle \scriptstyle \mu :=\nabla u{\mathcal {L}}}$

${\displaystyle \scriptstyle {\mathcal {L}}}$

${\displaystyle \Omega }$

${\displaystyle \int u\operatorname {div} \phi =-\int \phi \,d\mu =-\int \phi \,\nabla u\qquad \forall \phi \in C_{c}^{1}}$

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 ${\displaystyle W^{1,1}}$: 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 ${\displaystyle f}$ is a real function of bounded variation on an interval ${\displaystyle [a,b]}$ then

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

For real functions of several real variables

• the characteristic function of a Caccioppoli set is a BV function: BV functions lie at the basis of the modern theory of perimeters.
• Minimal surfaces are graphs of BV functions: in this context, see reference (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 (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.

• The
  minimization of such functional
  .
• Total variation denoising

See also

Notes