BOUNDED VARIATION

In mathematics, 'bounded variation' refers to a real-valued functions whose total variation is bounded i.e. is less than infinity: 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 y-axis (i.e. the distance calculated neglecting the contribution of motion along x-axis) traveled by an ideal point moving along the graph of the given function (which, under given hypothesis, is also a continuous path) has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is an hypersurface in this case), but can be every intersection of the graph itself with a plane parallel to a fixed x-axis and to the y-axis.
Functions of bounded variation are precisely those with respect to which one may find Riemann-Stieltjes integrals of all continuous functions.
Another characterization states that the functions of bounded variation on a closed interval are exactly those ''$f$'' which can be written as a difference ''$g-h$'', where both ''$g$'' and ''$h$'' are monotone.
In the case of several variables, a function ''$f$'' defined on an open subset of $scriptstylemathbb\{R\}^n$ is said to have bounded variation if its distributional derivative is a finite vector Radon measure.

Contents

History

Formal definition

''BV'' functions of one variable

''BV'' functions of several variables

Generalizations

Weighted ''BV'' functions

''SBV'' functions

Examples

Applications

Mathematics

Physics and engineering

See also

References

Bibliography

External links

Theory

Other

History

According to 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 step in the generalization of this concept to functions of several variables was due to Leonida Tonelli, 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 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 measure of non smooth boundaries of sets (see voice "''Caccioppoli set''" for further informations), Edward D. Conway and Joel A. Smoller applied them 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.

Formal definition

''BV'' functions of one variable

'Definition 1.' The 'total variation' of a real-valued function ''$f$'', defined on a interval $scriptstyle\; [a\; ,\; b]\; subset\; mathbb\{R\}$ is the quantity
:$V^a\_b(f)=sup\_\{P\; in\; mathcal\{P\}\}\; sum\_\{i=0\}^\{n\_P-1\}\; |\; f(x\_\{i+1\})-f(x\_i)\; |.\; ,$
where the supremum is taken over the set $scriptstyle\; mathcal\{P\}\; =left\{P=\{\; x\_0,\; dots\; ,\; x\_\{n\_p\}\}|P\; ext\{\; is\; a\; partition\; of\; \}\; [a,b]$
ight}
of all partitions of the interval considered.
If $f$ is differentiable and its derivative is integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,
:$V^a\_b(f)\; =\; int\; \_a^b\; |f\text{'}(x)|,\; dx.$
'Definition 2'. A real-valued function $f$ on the real line is said to be of 'bounded variation' ('BV function') on a chosen interval if its total variation is finite, ''i.e.''
:

''BV'' functions of several variables

Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure. More precisely:
'Definition 1' Let '$Omega$' be an open subset of $scriptstylemathbb\{R\}^n$. A function '$u$' in $L^1(Omega)$ is said of 'bounded variation' ('BV function'), and write
:$uin\; BV(Omega)$
if there exists a finite vector Radon measure $scriptstyle\; Duinmathcal\; M(Omega,mathbb\{R\}^n)$ such that the following equality holds
:$$
int_Omega u(x),mathrm{div}phi(x), dx = - int_Omega langle phi(x), Du(x)
angle
qquad orall phiin C_c^1(Omega,mathbb{R}^n)

that is, '$u$' defines a linear functional on the space $scriptstyle\; C\_c^1(Omega,mathbb\{R\}^n)$ of continuously differentiable vector functions $scriptstylephi$ of compact support contained in $Omega$: the vector measure '$Du$' represents therefore the distributional or weak gradient of $u$.
An equivalent definition is the following.
'Definition 2' Given a function '$u$' belonging to $scriptstyle\; L^1(Omega)$, the 'total variation of $u$' in $Omega$ is defined as
:$V(u,Omega):=supleft\{int\_Omega\; umathrm\{div\}phicolon\; phiin\; C\_c^1(Omega,mathbb\{R\}^n),\; Vert\; phiVert\_\{L^infty(Omega)\}le\; 1$
ight}.
where $scriptstyle\; Vert;Vert\_\{L^infty(Omega)\}$ is the essential supremum norm.
The space of 'functions of bounded variation' ('BV functions') can then be defined as
:
Notice that the Sobolev space $W^\{1,1\}(Omega)$ is a proper subset of $BV(Omega)$. In fact, for each '$u$' in $W^\{1,1\}(Omega)$ it is possible to choose a measure $scriptstyle\; mu:=$
abla u mathcal L (where $scriptstylemathcal\; L$ is the Lebesgue measure on $Omega$) such that the equality
:$int\; umathrm\{div\}phi\; =\; -int\; phi,\; dmu\; =\; -int\; phi$
abla u qquad orall phiin C_c^1
holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a ''BV'' function which is not $W^\{1,1\}.$

Generalizations

Weighted ''BV'' functions

It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let be any increasing function such that
ightarrow 0_+}arphi(x) = 0 (the 'weight function') and let $scriptstyle\; f\; :\; [0,\; T]longrightarrow\; X$ be a function from the interval $scriptstyle\; [0\; ,\; T]\; subset\; mathbb\{R\}$ taking values in a normed vector space $X$. Then the '-variation' of $f$ over $[0,\; T]$ is defined as
:
ight),
where, as usual, the supremum is taken over all finite partitions of the interval $[0,\; T]$, i.e. all the finite sets of real numbers $t\_i$ such that
:
The original notion of variation considered above is the special case of -variation for which the weight function is the identity function: therefore a integrable function $f$ is said to be a 'weighted ''BV'' function' ('of weight' ) if and only if its -variation is finite.
:
The space is a topological vector space with respect to the norm
:
where $scriptstyle|\; f\; |\_\{infty\}$ denotes the usual supremum norm of ''$f$''.

''SBV'' functions

'SBV functions' ''i.e.'' ''Special functions of Bounded Variation'' where introduced by Luigi Ambrosio and Ennio de Giorgi in the paper , dealing with free discontinuity variational problems: given a open subset '$Omega$' of $scriptstylemathbb\{R\}^n$, the space $scriptstyle\; \{SBV\}(Omega)$ is a proper subspace of $scriptstyle\; BV(Omega)$, since the weak gradient of each function belonging to it const exatcly of the sum of a $n$-dimensional support and a $n-1$-dimensional support measure and ''no lower-dimensional terms'', as seen in the following definition.
'Definition'. Given a function '$u$' belonging to $scriptstyle\; L^1(Omega)$, then $scriptstyle\; uin\; \{SBV\}(Omega)$ if and only if
'1.' There exist two Borel functions $f$ and $g$ of domain $Omega$ and Codomain $scriptstyle\; mathbb\{R\}^n$ such that
:
'2.' For all of continuously differentiable vector functions $scriptstylephi$ of compact support contained in $Omega$, ''i.e.'' for all $scriptstyle\; phi\; in$
C_c^1(Omega,mathbb{R}^n) the following formula is true:
:$int\_Omega\; umbox\{div\}\; phi\; dH^n\; =\; int\_Omega\; langle\; phi,\; f$
angle dH^n +int_Omega langle phi, g
angle dH^{n-1}.
where is the -dimensional Hausdorff measure.
Details on the properties of ''SBV'' functions can be found in works cited in the bibliography section: particularly the paper contains a useful bibliography.

Examples

The function
:
eq 0 end{cases}
is ''not'' of bounded variation on the interval $[0,\; 2/pi]$
While it is harder to see, the function
:
eq 0 end{cases}
is ''not'' of bounded variation on the interval $[0,\; 2/pi]$ either.
At the same time, the function
:
eq 0 end{cases}
''is'' of bounded variation on the interval $[0,2/pi]$.

Applications

Mathematics

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

★ $f$ is continuous except at most on a countable set;

★ $f$ has one-sided limits everywhere (limits from the left everywhere in $(a,b]$, and from the right everywhere in [''a'',''b'') );

★ the derivative $f\text{'}(x)$ exists almost everywhere (i.e. except for a set of measure zero).

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.

See also

★ Caccioppoli set

★ Ennio de Giorgi

★ Helly's selection theorem

★ Radon measure

★ Renato Caccioppoli

★ Riemann-Stieltjes integral

★ Total variation

★ Weak derivative

References

★ . Some recollections from one of the founders of the theory of ''BV'' functions of several variables (in Italian).

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

★ ISBN 90-247-3109-7. The whole book is devoted to the theory of ''BV'' functions and their applications to problems in mathematical physics involving discontinuous functions and geometric objects with non-smooth boundaries.

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

★ ISBN 3-7643-1907-0.

★ ISBN 0-486-66289-6.

★ .

Bibliography

★ . A paper containing a demonstration of the compactness of the set of SBV functions.

★ (in Italian). The first paper about ''SBV'' functions and related variational problems.

★ . An important paper where properties of ''BV'' functions were applied to ''single'' hyperbolic equations of first order.

★ . A survey paper on free-discontinuity variational problems including several details on the theory of ''SBV'' functions, their applications and a rich bibliography (in Italian), written by Ennio de Giorgi.

External links

Theory

★ Boris I. Golubov (and comments of Anatolii Georgievich Vitushkin) "''Variation of a function''", Springer-Verlag Online Encyclopaedia of Mathematics.

★ .

★ (at Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.

★ Rowland, Todd and Weisstein, Eric W. "''Bounded Variation''". From MathWorld--A Wolfram Web Resource.

Other

★ Luigi Ambrosio home page at the Scuola Normale Superiore, 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, Pisa.

★ Aizik Isaakovich Volpert at Technion. Academic home page of one of the leading contributors to the theory of ''BV'' functions.
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