In mathematical analysis, a function of bounded variation refers to 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 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 <math>f</math> which can be written as a difference <math>g-h</math>, where both <math>g</math> and <math>h</math> are monotone. In the case of several variables, a function <math>f</math> defined on an open subset of <math>\scriptstyle\mathbb{R}^n</math> is said to have bounded variation if its distributional derivative is a finite vector Radon measure.
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History
According to 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 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 (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 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 (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.
Formal definition
BV functions of one variable
Definition 1. The total variation of a real-valued function <math>f</math>, defined on a interval <math> \scriptstyle [a , b] \subset \mathbb{R}</math> is the quantity
- <math> V^a_b(f)=\sup_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |. \,</math>
where the supremum is taken over the set <math> \scriptstyle \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_p}\}|P\text{ is a partition of } [a,b] \right\} </math> of all partitions of the interval considered. If <math> f </math> 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,
- <math> V^a_b(f) = \int _a^b |f'(x)|\, dx.</math>
Definition 2. A real-valued function <math> f </math> on the real line is said to be of bounded variation (BV function) on a chosen interval <math> [a,b]</math> if its total variation is finite, i.e.
- <math> f \in BV([a,b]) \iff V^a_b(f) < +\infty </math>
It can be proved that a real function f is of bounded variation in an interval if and only if it can be written as the difference <math> f=f_1 - f_2</math> of two non decreasing functions (This is known as the Jordan decomposition.)
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 <math> \Omega </math> be an open subset of <math> \scriptstyle\mathbb{R}^n </math>. A function <math> u </math> in <math> L^1(\Omega) </math> is said of bounded variation (BV function), and write
- <math> u\in BV(\Omega)</math>
if there exists a finite vector Radon measure <math> \scriptstyle Du\in\mathcal M(\Omega,\mathbb{R}^n)</math> such that the following equality holds
- <math>
\int_\Omega u(x)\,\mathrm{div}\phi(x)\, dx = - \int_\Omega \langle \phi(x), Du(x)\rangle \qquad \forall \phi\in C_c^1(\Omega,\mathbb{R}^n) </math> that is, <math>u</math> defines a linear functional on the space <math> \scriptstyle C_c^1(\Omega,\mathbb{R}^n)</math> of continuously differentiable vector functions <math> \scriptstyle\phi </math> of compact support contained in <math> \Omega </math>: the vector measure <math>Du</math> represents therefore the distributional or weak gradient of <math>u</math>. An equivalent definition is the following. Definition 2 Given a function <math>u</math> belonging to <math> \scriptstyle L^1(\Omega)</math>, the total variation of <math>u</math> in <math>\Omega</math> is defined as
- <math> V(u,\Omega):=\sup\left\{\int_\Omega u\mathrm{div}\phi\colon \phi\in C_c^1(\Omega,\mathbb{R}^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\right\}. </math>
where <math> \scriptstyle \Vert\;\Vert_{L^\infty(\Omega)}</math> is the essential supremum norm. The space of functions of bounded variation (BV functions) can then be defined as
- <math> BV(\Omega)=\{ u\in L^1(\Omega)\colon V(u,\Omega)<+\infty\}</math>
Notice that the Sobolev space <math> W^{1,1}(\Omega)</math> is a proper subset of <math> BV(\Omega)</math>. In fact, for each <math> u </math> in <math> W^{1,1}(\Omega) </math> it is possible to choose a measure <math> \scriptstyle \mu:=\nabla u \mathcal L</math> (where <math> \scriptstyle\mathcal L</math> is the Lebesgue measure on <math>\Omega</math>) such that the equality
- <math> \int u\mathrm{div}\phi = -\int \phi\, d\mu = -\int \phi \nabla u \qquad \forall \phi\in C_c^1 </math>
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 <math>W^{1,1}.</math>
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 <math>\scriptstyle \varphi : [0, +\infty)\longrightarrow [0, +\infty)</math> be any increasing function such that <math>\scriptstyle \varphi(0) = \varphi(0+) =\lim_{x\rightarrow 0_+}\varphi(x) = 0</math> (the weight function) and let <math>\scriptstyle f : [0, T]\longrightarrow X </math> be a function from the interval <math> \scriptstyle [0 , T] \subset \mathbb{R}</math> taking values in a normed vector space <math>X</math>. Then the <math>\scriptstyle \boldsymbol\varphi</math>-variation of <math>f</math> over <math>[0, T]</math> is defined as
- <math>\mathop{\varphi\mbox{-Var}}_{[0, T]} (f) := \sup \sum_{j = 0}^{k} \varphi \left( | f(t_{j + 1}) - f(t_{j}) |_{X} \right),</math>
where, as usual, the supremum is taken over all finite partitions of the interval <math>[0, T]</math>, i.e. all the finite sets of real numbers <math>t_i</math> such that
- <math>0 = t_{0} < t_{1} < \ldots < t_{k} = T.</math>
The original notion of variation considered above is the special case of <math>\scriptstyle \varphi</math>-variation for which the weight function is the identity function: therefore a integrable function <math>f</math> is said to be a weighted BV function (of weight <math>\varphi</math>) if and only if its <math>\scriptstyle \varphi</math>-variation is finite.
- <math>f\in BV_\varphi([0, T];X)\iff \mathop{\varphi\mbox{-Var}}_{[0, T]} (f) <+\infty</math>
The space <math>\scriptstyle BV_\varphi([0, T];X)</math> is a topological vector space with respect to the norm
- <math>\| f \|_{BV_\varphi} := \| f \|_{\infty} + \mathop{\varphi \mbox{-Var}}_{[0, T]} (f),</math>
where <math>\scriptstyle\| f \|_{\infty}</math> denotes the usual supremum norm of <math>f</math>.
SBV functions
SBV functions i.e. Special functions of Bounded Variation where introduced by Luigi Ambrosio and Ennio de Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuity variational problems: given a open subset <math> \Omega </math> of <math> \scriptstyle\mathbb{R}^n </math>, the space <math>\scriptstyle {SBV}(\Omega)</math> is a proper subspace of <math>\scriptstyle BV(\Omega)</math>, since the weak gradient of each function belonging to it const exatcly of the sum of a <math>n</math>-dimensional support and a <math>n-1</math>-dimensional support measure and no lower-dimensional terms, as seen in the following definition. Definition. Given a function <math>u</math> belonging to <math> \scriptstyle L^1(\Omega)</math>, then <math>\scriptstyle u\in {SBV}(\Omega) </math> if and only if 1. There exist two Borel functions <math>f</math> and <math>g</math> of domain <math>\Omega</math> and codomain <math>\scriptstyle \mathbb{R}^n </math> such that
- <math> \int_\Omega\vert f\vert dH^n+ \int_\Omega\vert g\vert dH^{n-1}<+\infty.</math>
2. For all of continuously differentiable vector functions <math> \scriptstyle\phi </math> of compact support contained in <math> \Omega </math>, i.e. for all <math> \scriptstyle \phi \in C_c^1(\Omega,\mathbb{R}^n)</math> the following formula is true:
- <math> \int_\Omega u\mbox{div} \phi dH^n = \int_\Omega \langle \phi, f\rangle dH^n +\int_\Omega \langle \phi, g\rangle dH^{n-1}.</math>
where <math>H^\alpha</math> is the <math>\alpha</math>-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.
Examples
The function
- <math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>
is not of bounded variation on the interval <math> [0, 2/\pi]</math> While it is harder to see, the function
- <math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>
is not of bounded variation on the interval <math> [0, 2/\pi]</math> either. At the same time, the function
- <math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x^2 \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>
is of bounded variation on the interval <math> [0,2/\pi]</math>.
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 <math>f</math> is a real function of bounded variation on an interval [a, b] then
- <math>f</math> is continuous except at most on a countable set;
- <math>f</math> has one-sided limits everywhere (limits from the left everywhere in <math>(a,b]</math>, and from the right everywhere in [a,b) );
- the derivative <math>f'(x)</math> 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
- Renato Caccioppoli
- Riemann-Stieltjes integral
- Total variation
References
- Cesari, Lamberto (1986), "L'opera di Leonida Tonelli e la sua influenza nel pensiero scientifico del secolo (the work of Leonida Tonelli and his influence on scientific thinking in this century)", written at Rome, 6-9 June 1985, in G. Montalenti et als., Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli), Accademia Nazionale dei Lincei, Atti dei Convegni Lincei, Vol. 77, 41-73, <http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847>. Some recollections from one of the founders of the theory of BV functions of several variables (in Italian).
- Giusti, Enrico (1984), written at Basel, Minimal surfaces and functions of bounded variations, Birkhäuser Verlag ISBN 0-8176-3153-4, particularly part I, chapter 1 "Functions of bounded variation and Caccioppoli sets".
- Hudjaev, Sergei Ivanovich & Aizik Isaakovich Vol'pert (1986), written at Dordrecht, Analysis in classes of discontinuous functions and equation of mathematical physics, Martinus Nijhoff Publishers 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.
- Kannan, Rangachary & Carole King Krueger (1996), written at Berlin-Heidelberg-New York, Advanced analysis on the real line, Springer Verlag 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
- Moreau, J.J. (1988), "Bounded variation in time.", written at Basel, in Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G., Topics in nonsmooth mechanics., Birkhäuser Verlag, 1-74 ISBN 3-7643-1907-0.
- Riesz, Frigyes & Bela Sz-Nagy (1990), written at New York, Functional Analysis, Dover Publications ISBN 0-486-66289-6.
- Vol'pert, Aizik Isaakovich (1967), "Spaces BV and quasi-linear equations", Math.Sbornik-USSR 2: 225-267.
Bibliography
- Alberti, Giovanni & Carlo Mantegazza (1997), "A note on the theory of SBV functions", Bollettino Unione Matematica Italiana, Sezione B 7: 375-382. A paper containing a demonstration of the compactness of the set of SBV functions.
- Ambrosio, Luigi & Ennio De Giorgi (1988), "Un nuovo tipo di funzionale del calcolo delle variazioni (A new kind of functional in the calculus of variations)", Atti dell'Accademia Nazionale dei Lincei, Rendiconti Lincei, Classe di Scienze Fisiche, Mathematiche, Naturali 82: 199-210 (in Italian). The first paper about SBV functions and related variational problems.
- Conway, Edward D & Joel A. Smoller (1966), "Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables", Communications on Pure and Applied Mathematics 19: 95-105. An important paper where properties of BV functions were applied to single hyperbolic equations of first order.
- De Giorgi, Ennio (1992), "Problemi variazionali con discontinuità libere (Free-discontinuity variational problems)", written at Roma, 8-11 October 1990, in E. Amaldi et als., Convegno internazionale in memoria di Vito Volterra (International congress in memory of Vito Volterra), Accademia Nazionale dei Lincei, Atti dei Convegni Lincei, Vol. 92, 133-150. 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.
- BV function on PlanetMath.
- Jordan, Camille (1881), written at Paris, janv.-juin, "Sur la série de Fourier", Comptes rendus des Académie des sciences de Paris 92: 228-230 (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.
This article incorporates material from BV function on PlanetMath, which is licensed under the GFDL.
