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Bounded variation

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 compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many.

In the case of several variables, a function f defined on an open subset Ωof${\displaystyle \mathbb {R} ^{n}}$ is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure.

One of the most important aspects of functions of bounded variation is that they form an algebraofdiscontinuous 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 equationsinmathematics, physics and engineering.

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 ⊆ differentiable almost everywhere

History[edit]

According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper (Jordan 1881) dealing with the convergence of Fourier series. The first successful step in the generalization of this concept to functions of several variables was due to Leonida Tonelli,^[1] who introduced a class of continuous BV functions in 1926 (Cesari 1986, pp. 47–48), to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in (Cesari 1936), Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions 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 measureofnonsmooth boundariesofsets (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 (Oleinik 1957), and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper (Oleinik 1959): few years later, Edward D. Conway and Joel A. Smoller applied BV-functions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper (Conway & Smoller 1966), proving that the solution of the Cauchy 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 (Vol'pert 1967) he proved the chain rule for BV functions and in the book (Hudjaev & Vol'pert 1985) he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper (Ambrosio & Dal Maso 1990).

Formal definition[edit]

BV functions of one variable[edit]

Definition 1.1. The total variation of a real-valued (or more generally complex-valued) function f, defined on an interval ${\displaystyle [a,b]\subset \mathbb {R} }$ is the quantity

${\displaystyle V_{a}^{b}($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 ${\textstyle {\mathcal {P}}=\left\{P=\{x_{0},\dots ,x_{n_{P}}\}\mid P{\text{ is a partition of }}[a,b]{\text{ satisfying }}x_{i}\leq x_{i+1}{\text{ for }}0\leq i\leq n_{P}-1\right\}}$ of all partitions of the interval considered.

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

${\displaystyle V_{a}^{b}($f)=\int _{a}^{b}|f'(x)|\,\mathrm {d} x.}

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

${\displaystyle f\in \operatorname {BV} ([a,b])\iff V_{a}^{b}($f)<+\infty }

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 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 ${\displaystyle [a,b]}$ defines a bounded linear functionalon${\displaystyle C([a,b])}$. In this special case,^[2] the Riesz–Markov–Kakutani representation theorem states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in spectral theory,^[3] in particular in its application to ordinary differential equations.

BV functions of several variables[edit]

Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite^[4] Radon measure. More precisely:

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

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

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

${\displaystyle \int _{\Omega }u($x)\operatorname {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})}

that is, ${\displaystyle u}$ defines a linear functional on the space ${\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$ofcontinuously differentiable vector functions ${\displaystyle {\boldsymbol {\phi }}}$ofcompact support contained in ${\displaystyle \Omega }$: the vector measure ${\displaystyle Du}$ represents therefore the distributionalorweak gradientof${\displaystyle u}$.

BV can be defined equivalently in the following way.

Definition 2.2. Given a function ${\displaystyle u}$ belonging to ${\displaystyle L^{1}(\Omega )}$, the total variation of ${\displaystyle u}$^[5]in${\displaystyle \Omega }$ is defined as

${\displaystyle V(u,\Omega ):=\sup \left\{\int _{\Omega }u($x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x:{\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n}),\ \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1\right\}}

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

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

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

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

${\displaystyle \operatorname {\operatorname {BV} } (\Omega )=\{u\in L^{1}(\Omega )\colon V(u,\Omega )<+\infty \}}$

The two definitions are equivalent since if ${\displaystyle V(u,\Omega )<+\infty }$ then

${\displaystyle \left|\int _{\Omega }u($x)\operatorname {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})}

therefore ${\textstyle \displaystyle {\boldsymbol {\phi }}\mapsto \,\int _{\Omega }u($x)\operatorname {div} {\boldsymbol {\phi }}(x)\,dx} defines a continuous linear functional on the space ${\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$. Since ${\displaystyle C_{c}^{1}(\Omega ,\mathbb {R} ^{n})\subset C^{0}(\Omega ,\mathbb {R} ^{n})}$ as a linear subspace, this continuous linear functional can be extended continuously and linearly to the whole ${\displaystyle C^{0}(\Omega ,\mathbb {R} ^{n})}$ by the Hahn–Banach theorem. Hence the continuous linear functional defines a Radon measure by the Riesz–Markov–Kakutani representation theorem.

Locally BV functions[edit]

If the function spaceoflocally integrable functions, i.e. functions belonging to ${\displaystyle L_{\text{loc}}^{1}(\Omega )}$, is considered in the preceding definitions 1.2, 2.1 and 2.2 instead of the one of globally integrable functions, then the function space defined is that of functions of locally bounded variation. Precisely, developing this idea for definition 2.2, a local variation is defined as follows,

${\displaystyle V(u,U):=\sup \left\{\int _{\Omega }u($x)\operatorname {div} {\boldsymbol {\phi }}(x)\,\mathrm {d} x:{\boldsymbol {\phi }}\in C_{c}^{1}(U,\mathbb {R} ^{n}),\ \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1\right\}}

for every set ${\displaystyle U\in {\mathcal {O}}_{c}(\Omega )}$, having defined ${\displaystyle {\mathcal {O}}_{c}(\Omega )}$ as the set of all precompact open subsetsof${\displaystyle \Omega }$ with respect to the standard topologyoffinite-dimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as

${\displaystyle \operatorname {BV} _{\text{loc}}(\Omega )=\{u\in L_{\text{loc}}^{1}(\Omega )\colon \,(\forall U\in {\mathcal {O}}_{c}(\Omega ))\,V(u,U)<+\infty \}}$

Notation[edit]

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

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

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

Basic properties[edit]

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[edit]

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 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 continuumofdirections along which it is possible to approach a given point ${\displaystyle x_{0}}$ belonging to the domain ${\displaystyle \Omega }$⊂${\displaystyle \mathbb {R} ^{n}}$. It is necessary to make precise a suitable concept of limit: choosing a unit vector ${\displaystyle {\boldsymbol {\hat {a}}}\in \mathbb {R} ^{n}}$ it is possible to divide ${\displaystyle \Omega }$ in two sets

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

Then for each point ${\displaystyle x_{0}}$ belonging to the domain ${\displaystyle \Omega \in \mathbb {R} ^{n}}$ 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 subsetof${\displaystyle \Omega }$ 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)[edit]

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

${\displaystyle {\begin{aligned}\liminf _{n\rightarrow \infty }V(u_{n},\Omega )&\geq \liminf _{n\rightarrow \infty }\int _{\Omega }u_{n}($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\end{aligned}}}

Now considering the supremum on the set of functions ${\displaystyle {\boldsymbol {\phi }}\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$ such that ${\displaystyle \Vert {\boldsymbol {\phi }}\Vert _{L^{\infty }(\Omega )}\leq 1}$ then the following inequality holds true

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

which is exactly the definition of lower semicontinuity.

BV(Ω) is a Banach space[edit]

By definition ${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$ is a subsetof${\displaystyle L^{1}(\Omega )}$, 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 ${\displaystyle \phi \in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$ therefore ${\displaystyle u+v\in \operatorname {\operatorname {BV} } (\Omega )}$for all ${\displaystyle u,v\in \operatorname {\operatorname {BV} } (\Omega )}$, 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 ${\displaystyle cu\in \operatorname {\operatorname {BV} } (\Omega )}$ for all ${\displaystyle u\in \operatorname {\operatorname {BV} } (\Omega )}$, and all ${\displaystyle c\in \mathbb {R} }$. The proved vector space properties imply that ${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$ is a vector subspaceof${\displaystyle L^{1}(\Omega )}$. Consider now the function ${\displaystyle \|\;\|_{\operatorname {BV} }:\operatorname {\operatorname {BV} } (\Omega )\rightarrow \mathbb {R} ^{+}}$ defined as

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

where ${\displaystyle \|\;\|_{L^{1}}}$ is the usual ${\displaystyle L^{1}(\Omega )}$ norm: it is easy to prove that this is a normon${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$. To see that ${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$iscomplete respect to it, i.e. it is a Banach space, consider a Cauchy sequence ${\displaystyle \{u_{n}\}_{n\in \mathbb {N} }}$in${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$. By definition it is also a Cauchy sequencein${\displaystyle L^{1}(\Omega )}$ and therefore has a limit ${\displaystyle u}$in${\displaystyle L^{1}(\Omega )}$: since ${\displaystyle u_{n}}$ is bounded in ${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$ for each ${\displaystyle n}$, then ${\displaystyle \Vert u\Vert _{\operatorname {BV} }<+\infty }$bylower semicontinuity of the variation ${\displaystyle V(\cdot ,\Omega )}$, therefore ${\displaystyle u}$ is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number ${\displaystyle \varepsilon }$

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

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

BV(Ω) is not separable[edit]

To see this, it is sufficient to consider the following example belonging to the space ${\displaystyle \operatorname {\operatorname {BV} } ([0,1])}$:^[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 ${\displaystyle \alpha ,\beta \in [0,1]}$ such that ${\displaystyle \alpha \neq \beta }$ the following relation holds true:

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

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

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

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

Chain rule for BV functions[edit]

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

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

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

where ${\displaystyle {\bar {f}}({\boldsymbol {u}}({\boldsymbol {x}}))}$ is the mean value of the function at the point ${\displaystyle x\in \Omega }$, 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 ${\displaystyle f:\mathbb {R} ^{p}\rightarrow \mathbb {R} ^{s}}$ 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 'BV' function and choosing ${\displaystyle f((u,v))=uv}$, the preceding formula gives the Leibniz rule for '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 ${\displaystyle \operatorname {\operatorname {BV} } (\Omega )}$ is an algebra.

BV(Ω) is a Banach algebra[edit]

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