In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the -axis, neglecting the contribution of motion along -axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed -axis and to the -axis.
Functions of bounded variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions.
Another characterization states that the functions of bounded variation on a compact interval are exactly those which can be written as a difference, where both and are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many.
In the case of several variables, a function defined on an open subset of
Rn
One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering.
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
According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper 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, to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in, Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of two variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics. Renato Caccioppoli and Ennio De Giorgi used them to define measure of nonsmooth boundaries of sets (see the entry "Caccioppoli set" for further information). Olga Arsenievna Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper, and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper : 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, 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 he proved the chain rule for BV functions and in the book 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 .
[a,b]\subsetR
b(f)=\sup | |
V | |
P\inl{P |
where the supremum is taken over the set of all partitions of the interval considered.
If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,
b(f) | |
V | |
a |
=\int
b | |
a |
|f'(x)|dx.
A real-valued function
f
[a,b]\subsetR
f\in\operatorname{BV}([a,b])\iff
b(f) | |
V | |
a |
<+infty
It can be proved that a real function
f
[a,b]
f=f1-f2
f1
f2
[a,b]
Through the Stieltjes integral, any function of bounded variation on a closed interval
[a,b]
C([a,b])
Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite[4] Radon measure. More precisely:
Let
\Omega
Rn
u
L1(\Omega)
u\in\operatorname\operatorname{BV}(\Omega)
Du\inlM(\Omega,Rn)
\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx=-\int\Omega\langle\boldsymbol{\phi},Du(x)\rangle \forall\boldsymbol{\phi}\in
1(\Omega,R | |
C | |
c |
n)
that is,
u
1(\Omega,R | |
C | |
c |
n)
\boldsymbol{\phi}
\Omega
Du
u
BV can be defined equivalently in the following way.
Given a function
u
L1(\Omega)
u
\Omega
V(u,\Omega):=\sup\left\{\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx:\boldsymbol{\phi}\in
1(\Omega,R | |
C | |
c |
n), \Vert\boldsymbol{\phi}\Vert | |
Linfty(\Omega) |
\le1\right\}
where
\Vert \Vert | |
Linfty(\Omega) |
\int\Omega\vertDu\vert=V(u,\Omega)
in order to emphasize that
V(u,\Omega)
u
u
C1
The space of functions of bounded variation (BV functions) can then be defined as
\operatorname\operatorname{BV}(\Omega)=\{u\inL1(\Omega)\colonV(u,\Omega)<+infty\}
The two definitions are equivalent since if
V(u,\Omega)<+infty
\left|\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx\right|\leq
V(u,\Omega)\Vert\boldsymbol{\phi}\Vert | |
Linfty(\Omega) |
\forall\boldsymbol{\phi}\in
1(\Omega,R | |
C | |
c |
n)
therefore defines a continuous linear functional on the space
1(\Omega,R | |
C | |
c |
n)
1(\Omega,R | |
C | |
c |
n) \subsetC0(\Omega,Rn)
C0(\Omega,Rn)
If the function space of locally integrable functions, i.e. functions belonging to
1 | |
L | |
loc(\Omega) |
V(u,U):=\sup\left\{\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx:\boldsymbol{\phi}\in
1(U,R | |
C | |
c |
n), \Vert\boldsymbol{\phi}\Vert | |
Linfty(\Omega) |
\le1\right\}
U\inl{O}c(\Omega)
l{O}c(\Omega)
\Omega
\operatorname{BV}loc(\Omega)=\{u\in
1 | |
L | |
loc(\Omega)\colon |
(\forallU\inl{O}c(\Omega))V(u,U)<+infty\}
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 (partially), (partially), and is the following one
\operatorname\operatorname{BV}(\Omega)
\operatorname\operatorname{BV}loc(\Omega)
\overline{\operatorname\operatorname{BV}}(\Omega)
\operatorname\operatorname{BV}(\Omega)
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, and are extensively used.
In the case of one variable, the assertion is clear: for each point
x0
[a,b]\subsetR
u
\lim | |||||||
|
u(x)=
\lim | |||||||
|
u(x)
\lim | |||||||
|
u(x) ≠
\lim | |||||||
|
u(x)
while both limits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a continuum of directions along which it is possible to approach a given point
x0
\Omega
Rn
{\boldsymbol{\hat{a}}}\inRn
\Omega
\Omega({\boldsymbol{\hat{a
Then for each point
x0
\Omega\inRn
u
\lim\overset{\boldsymbol{x → \boldsymbol{x}0}{\boldsymbol{x}\in\Omega({\boldsymbol{\hat{a
\lim\overset{\boldsymbol{x → \boldsymbol{x}0}{\boldsymbol{x}\in\Omega({\boldsymbol{\hat{a
or
x0
\Omega
n-1
\lim\overset{\boldsymbol{x → \boldsymbol{x}0}{\boldsymbol{x}\in\Omega({\boldsymbol{\hat{a
are called approximate limits of the BV function
u
x0
V( ⋅ ,\Omega):\operatorname\operatorname{BV}(\Omega) → R+
\{un\}n\inN
u\in
1 | |
L | |
loc(\Omega) |
\begin{align} \liminfn → inftyV(un,\Omega)&\geq\liminfn → infty\int\Omegaun(x)\operatorname{div}\boldsymbol{\phi}dx\\ &\geq\int\Omega\limn → inftyun(x)\operatorname{div}\boldsymbol{\phi}dx\\ &=\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}dx \forall\boldsymbol{\phi}\in
1(\Omega,R | |
C | |
c |
n), \Vert\boldsymbol{\phi}\Vert | |
Linfty(\Omega) |
\leq1\end{align}
Now considering the supremum on the set of functions
\boldsymbol{\phi}\in
1(\Omega,R | |
C | |
c |
n)
\Vert\boldsymbol{\phi}\Vert | |
Linfty(\Omega) |
\leq1
\liminfn → inftyV(un,\Omega)\geqV(u,\Omega)
which is exactly the definition of lower semicontinuity.
By definition
\operatorname\operatorname{BV}(\Omega)
L1(\Omega)
\begin{align} \int\Omega[u(x)+v(x)]\operatorname{div}\boldsymbol{\phi}(x)dx&= \int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx+\int\Omegav(x)\operatorname{div}\boldsymbol{\phi}(x)dx=\\ &=-\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{align}
for all
\phi\in
1(\Omega,R | |
C | |
c |
n)
u+v\in\operatorname\operatorname{BV}(\Omega)
u,v\in\operatorname\operatorname{BV}(\Omega)
\int\Omegac ⋅ u(x)\operatorname{div}\boldsymbol{\phi}(x)dx= c\int\Omegau(x)\operatorname{div}\boldsymbol{\phi}(x)dx= -c\int\Omega\langle\boldsymbol{\phi}(x),Du(x)\rangle
for all
c\inR
cu\in\operatorname\operatorname{BV}(\Omega)
u\in\operatorname\operatorname{BV}(\Omega)
c\inR
\operatorname\operatorname{BV}(\Omega)
L1(\Omega)
\| \|\operatorname{BV
\|u\|\operatorname{BV
where
\|
\| | |
L1 |
L1(\Omega)
\operatorname\operatorname{BV}(\Omega)
\operatorname\operatorname{BV}(\Omega)
\{un\}n\inN
\operatorname\operatorname{BV}(\Omega)
L1(\Omega)
u
L1(\Omega)
un
\operatorname\operatorname{BV}(\Omega)
n
\Vertu\Vert\operatorname{BV
V( ⋅ ,\Omega)
u
\varepsilon
\Vertuj-uk\Vert\operatorname{BV
V( ⋅ ,\Omega)
To see this, it is sufficient to consider the following example belonging to the space
\operatorname\operatorname{BV}([0,1])
\chi\alpha=\chi[\alpha,1]= \begin{cases}0&ifx\notin [\alpha,1]\ 1&ifx\in[\alpha,1] \end{cases}
[\alpha,1]
\alpha,\beta\in[0,1]
\alpha\ne\beta
\Vert\chi\alpha-\chi\beta\Vert\operatorname{BV
\operatorname\operatorname{BV}(]0,1[)
\alpha\in[0,1]
B\alpha=\left\{\psi\in\operatorname\operatorname{BV}([0,1]);\Vert\chi\alpha-\psi\Vert\operatorname{BV
[0,1]
\operatorname\operatorname{BV}([0,1])
\operatorname{BV}loc
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 functions or functionals with a very limited degree of smoothness. The following chain rule is proved in the paper . Note all partial derivatives must be interpreted in a generalized sense, i.e., as generalized derivatives.
Theorem. Let
f:Rp → R
C1
\boldsymbol{u}(\boldsymbol{x})=(u1(\boldsymbol{x}),\ldots,up(\boldsymbol{x}))
\operatorname\operatorname{BV}(\Omega)
\Omega
Rn
f\circ\boldsymbol{u}(\boldsymbol{x})=f(\boldsymbol{u}(\boldsymbol{x}))\in\operatorname\operatorname{BV}(\Omega)
\partialf(\boldsymbol{u | |
(\boldsymbol{x}))}{\partial |
xi}=\sum
| ||||
k=1 |
u | ||||
|
xi} \foralli=1,\ldots,n
where
\barf(\boldsymbol{u}(\boldsymbol{x}))
x\in\Omega
\barf(\boldsymbol{u}(\boldsymbol{x}))=
1 | |
\int | |
0 |
f\left(\boldsymbol{u}\boldsymbol{\hat
f:Rp → Rs
(u(\boldsymbol{x}),v(\boldsymbol{x}))
\boldsymbolu(\boldsymbol{x})
v(\boldsymbol{x})
f((u,v))=uv
\partialv(\boldsymbol{x | |
)u(\boldsymbol{x})}{\partial |
xi}={\baru(\boldsymbol{x})}
\partialv(\boldsymbol{x | |
)}{\partial |
xi}+ {\barv(\boldsymbol{x})}
\partialu(\boldsymbol{x | |
)}{\partial |
xi}
This implies that the product of two functions of bounded variation is again a function of bounded variation, therefore
\operatorname\operatorname{BV}(\Omega)
This property follows directly from the fact that
\operatorname\operatorname{BV}(\Omega)
\{vn\}
\{un\}
v
u
\operatorname\operatorname{BV}(\Omega)
\begin{matrix} vun\xrightarrow[n\toinfty]{}vu\\ vnu\xrightarrow[n\toinfty]{}vu \end{matrix} \Longleftrightarrow vu\in\operatorname\operatorname{BV}(\Omega)
therefore the ordinary product of functions is continuous in
\operatorname\operatorname{BV}(\Omega)
It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let
\varphi:[0,+infty)\longrightarrow[0,+infty)
\varphi(0)=\varphi(0+)
=\lim | |
x → 0+ |
\varphi(x)=0
f:[0,T]\longrightarrowX
[0,T]
\subsetR
X
\boldsymbol\varphi
f
[0,T]
\varphi-\operatorname{Var
where, as usual, the supremum is taken over all finite partitions of the interval
[0,T]
ti
0=t0<t1< … <tk=T.
The original notion of variation considered above is the special case of
\varphi
f
\varphi
\varphi
f\in\operatorname{BV}\varphi([0,T];X)\iff\varphi-\operatorname{Var
The space
\operatorname{BV}\varphi([0,T];X)
\|f\|\operatorname{BV\varphi}:=\|f\|infty+\varphi-\operatorname{Var
where
\|f\|infty
f
\varphi(x)=xp
p
SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio De Giorgi in the paper, dealing with free discontinuity variational problems: given an open subset
\Omega
Rn
\operatorname{SBV}(\Omega)
\operatorname\operatorname{BV}(\Omega)
n
n-1
Definition. Given a locally integrable function
u
u\in\operatorname{SBV}(\Omega)
1. There exist two Borel functions
f
g
\Omega
Rn
\int\Omega\vertf\vertdHn+\int\Omega\vertg\vertdHn-1<+infty.
\phi
\Omega
\phi
1(\Omega,R | |
\in C | |
c |
n)
\int\Omegau\operatorname{div}\phidHn=\int\Omega\langle\phi,f\rangledHn+\int\Omega\langle\phi,g\rangledHn-1.
where
H\alpha
\alpha
Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper contains a useful bibliography.
As particular examples of Banach spaces, consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x = (xi) of real or complex numbers is defined by
\operatorname{TV}(x)=
infty | |
\sum | |
i=1 |
|xi+1-xi|.
The space of all sequences of finite total variation is denoted by BV. The norm on BV is given by
\|x\|\operatorname{BV
\ell1
The total variation itself defines a norm on a certain subspace of BV, denoted by BV0, consisting of sequences x = (xi) for which
\limn\toinftyxn=0.
\|x\|\operatorname{BV0}=\operatorname{TV}(x)=
infty | |
\sum | |
i=1 |
|xi+1-xi|.
\ell1
\mu
(X,\Sigma)
\Vert\mu\Vert=|\mu|(X)
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
f(x)=\begin{cases}0,&ifx=0\ \sin(1/x),&ifx ≠ 0\end{cases}
is not of bounded variation on the interval
[0,2/\pi]
While it is harder to see, the continuous function
f(x)=\begin{cases}0,&ifx=0\ x\sin(1/x),&ifx ≠ 0\end{cases}
is not of bounded variation on the interval
[0,2/\pi]
At the same time, the function
f(x)=\begin{cases}0,&ifx=0\ x2\sin(1/x),&ifx ≠ 0\end{cases}
is of bounded variation on the interval
[0,2/\pi]
[a,b]
a>0
Every monotone, bounded function is of bounded variation. For such a function
f
[a,b]
P=\{x0,\ldots,x
nP |
\}
nP-1 | |
\sum | |
i=0 |
|f(xi+1)-f(xi)|=|f(b)-f(a)|
from the fact that the sum on the left is telescoping. From this, it follows that for such
f
b(f)=|f(b)-f(a)|. | |
V | |
a |
In particular, the monotone Cantor function is a well-known example of a function of bounded variation that is not absolutely continuous.[8]
The Sobolev space
W1,1(\Omega)
\operatorname\operatorname{BV}(\Omega)
u
W1,1(\Omega)
\mu:=\nablaulL
lL
\Omega
\intu\operatorname{div}\phi=-\int\phid\mu=-\int\phi\nablau \forall\phi\in
1 | |
C | |
c |
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
W1,1
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
[a,b]
f
f
(a,b]
[a,b)
f'(x)
For real functions of several real variables
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 details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.