Bounded variation


Bounded variation

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 "f" which can be written as a difference "g-h", where both "g" and "h" are bounded monotone.

In the case of several variables, a function "f" defined on an open subset Omega of scriptstylemathbb{R}^n is said to have bounded variation if its distributional derivative is a finite vector Radon measure.

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. Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations on generalized functions, "function of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication".

History

According to Golubov, "BV" functions of a single variable were first introduced by Camille Jordan, in the paper Harv|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 Harv|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 Harv|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 Harv|Vol'pert|1967 and in the book Harv|Hudjaev|Vol'pert|1986.

Formal definition

"BV" functions of one variable

Definition 1. The total variation of a real-valued function "f", defined on an 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'(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 [a,b] if its total variation is finite, "i.e.": f in BV( [a,b] ) iff V^a_b(f) < +infty

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 f=f_1 - f_2 of two non decreasing functions (This is known as the Jordan decomposition.)

Through the Stieltjes integral, any function of bounded variation on a closed interval [a,b] defines a bounded linear functional on C( [a,b] ). In this special case harv|Kolmogorov|Fomin|1969|pp=374-376, the Riesz representation theorem states that every bounded linear functional arises uniquely in this way. The normalised positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in
spectral theory harv|Riesz|Sz.-Nagy|1990, in particular in its application to ordinary differential equations.

"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 locally integrable function u 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}oldsymbol{phi}(x), dx = - int_Omega langleoldsymbol{phi}, Du(x) angle qquad foralloldsymbol{phi}in 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 scriptstyleoldsymbol{phi} 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 locally integrable function u, the total variation of u in is defined as

: V(u,Omega):=supleft{int_Omega umathrm{div}oldsymbol{phi}colon phiin C_c^1(Omega,mathbb{R}^n), Vertoldsymbol{phi}Vert_{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

: BV(Omega)={ uin L^1(Omega)colon V(u,Omega)<+infty}

The two definition are equivalent since if scriptstyle V(u,Omega)<+infty then

:left|int_Omega u(x),mathrm{div}oldsymbol{phi}, dx ight |leq V(u,Omega)Vertoldsymbol{phi}Vert_{L^infty(Omega)}qquad forall oldsymbol{phi}in C_c^1(Omega,mathbb{R}^n)

therefore scriptstyle int_Omega u(x),mathrm{div}oldsymbol{phi}(x) defines a continuous linear functional on the space scriptstyle C_c^1(Omega,mathbb{R}^n). Since scriptstyle 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 linearily to the whole scriptstyle C^0(Omega,mathbb{R}^n) by the Hahn–Banach theorem i.e. it defines a Radon measure.

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. References Harv|Giusti|1984|pp=7-9, Harv|Hudjaev|Vol'pert|1986 and Harv|Màlek|Nečas|Rokyta|Růžička|1996 are extensively used.

"BV" functions have only jump-type singularities

In the case of one variable, the assertion is clear: for each point x_0 in the interval scriptstyle ] a , b [ subset mathbb{R} of definition of the function u, either one of the following two assertions is true

: lim_{x ightarrow x_{0^-!!!u(x) = !!!lim_{x ightarrow x_{0^+!!!u(x) : lim_{x ightarrow x_{0^-!!!u(x) eq !!!lim_{x ightarrow x_{0^+!!!u(x)

while both limits exists and are finite. In the case of functions 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 x_0 belonging to the domain scriptstyleOmegainmathbb{R}^n. It is necessary to make precise a suitable concept of limit: choosing a unit vector scriptstyle{oldsymbolhat a}inmathbb{R}^n it is possible to divide Omega in two sets

:Omega_{({oldsymbolhat a},oldsymbol{x}_0)} = Omega cap {oldsymbol{x}inmathbb{R}^n|langleoldsymbol{x}-oldsymbol{x}_0,{oldsymbolhat a} angle>0} qquad Omega_{(-{oldsymbolhat a},oldsymbol{x}_0)} = Omega cap {oldsymbol{x}inmathbb{R}^n|langleoldsymbol{x}-oldsymbol{x}_0,-{oldsymbolhat a} angle>0}

Then for each point x_0 belonging to the domain scriptstyleOmegainmathbb{R}^n of the "BV" function u or one of the following two assertion is true

: lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{({oldsymbolhat a},oldsymbol{x}_0)}!!!!!!u(oldsymbol{x}) = !!!!!!!lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{(-{oldsymbolhat a},oldsymbol{x}_0)}!!!!!!!u(oldsymbol{x}): lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{({oldsymbolhat a},oldsymbol{x}_0)}!!!!!!u(oldsymbol{x}) eq !!!!!!!lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{(-{oldsymbolhat a},oldsymbol{x}_0)}!!!!!!!u(oldsymbol{x})

or x_0 belongs to a subset of Omega having zero n-1-dimensional Hausdorff measure. The quantities

:lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{({oldsymbolhat a},oldsymbol{x}_0)}!!!!!!u(oldsymbol{x})=u_{oldsymbol{hat a(oldsymbol{x}) qquad lim_{overset{oldsymbol{x} ightarrow oldsymbol{x}_0}{oldsymbol{x}inOmega_{(-{oldsymbolhat a},oldsymbol{x}_0)}!!!!!!!u(oldsymbol{x})=u_{-oldsymbol{hat a(oldsymbol{x})

are called approximate limits of the "BV" function u at the point x_0.

"V"(&middot;, &Omega;) is lower semi-continuous on BV(&Omega;)

The functional scriptstyle V(cdot,Omega):BV(Omega) ightarrow mathbb{R}^+ is lower semi-continuous: to see this, choose a Cauchy sequence of "BV"-functions scriptstyle{u_n}_{ninmathbb{N converging to scriptstyle uin L^1_{loc}(Omega). Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit

:liminf_{n ightarrowinfty}V(u_n,Omega)geqlim_{n ightarrowinfty}int_Omega u_n(x),mathrm{div}oldsymbol{phi}, dx = int_Omega u(x),mathrm{div}oldsymbol{phi}, dx qquadforalloldsymbol{phi}in C_c^1(Omega,mathbb{R}^n),quadVertoldsymbol{phi}Vert_{L^infty(Omega)}leq 1

Now considering the supremum on the set of functions scriptstyleoldsymbol{phi}in C_c^1(Omega,mathbb{R}^n) such that scriptstyle Vertoldsymbol{phi}Vert_{L^infty(Omega)}leq 1 then the following inequality holds true

:liminf_{n ightarrowinfty}V(u_n,Omega)geq V(u,Omega)

which is exactly the definition of lower semicontinuity.

BV(&Omega;) is a Banach space

By definition BV(Omega) is a subset of "L""1""loc"(&Omega;), while linearity follows from the linearity properties of the defining integral i.e.

:egin{align}int_Omega [u(x)+v(x)] ,mathrm{div}oldsymbol{phi}(x) & =int_Omega u(x),mathrm{div}oldsymbol{phi}(x)+int_Omega v(x),mathrm{div}oldsymbol{phi}(x) = \ & =- int_Omega langleoldsymbol{phi}(x), Du(x) angle- int_Omega langle oldsymbol{phi(x)}, Dv(x) angle =- int_Omega langle oldsymbol{phi}(x), [Du(x)+Dv(x)] angle end{align}

for all scriptstylephiin C_c^1(Omega,mathbb{R}^n) therefore scriptstyle u+vin BV(Omega)for all scriptstyle u,vin BV(Omega), and

:int_Omega cu(x),mathrm{div}oldsymbol{phi}(x)=cint_Omega u(x),mathrm{div}oldsymbol{phi}(x)=-c int_Omega langle oldsymbol{phi}(x), Du(x) angle

for all scriptstyle cinmathbb{R}, therefore scriptstyle cuin BV(Omega) for all scriptstyle uin BV(Omega), and all scriptstyle cinmathbb{R}. The proved vector space properties imply that BV(Omega) is a vector subspace of L^1(Omega). Consider now the function scriptstyle|;|_{BV}:BV(Omega) ightarrowmathbb{R}^+ defined as

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

where scriptstyle| ; |_{L^1} is the usual L^1(Omega) norm: it is easy to prove that this is a norm on BV(Omega). To see that BV(Omega) is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence scriptstyle{u_n}_{ninmathbb{R in BV(Omega). By definition it is also a Cauchy sequence in L^1(Omega) and therefore has a limit u in L^1(Omega): since u_n is bounded in BV(Omega) for each n, then scriptstyle Vert u Vert_{BV} < +infty by lower semicontinuity of the variation scriptstyle V(cdot,Omega), therefore u is a "BV" function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number scriptstylevarepsilon

:Vert u_j - u_k Vert_{BV}

Chain rule for "BV" functions

Chain rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behavior is described by functions or functionals with a very limited degree of smoothness.The following version is proved in the paper Harv|Vol'pert|1967|p=248: all partial derivatives must be intended in a generalized sense. i.e. as generalized derivatives

Theorem. Let scriptstyle f:mathbb{R}^p ightarrowmathbb{R} be a function of class C^1 (i.e. a continuous and differentiable function having continuous derivatives) and let scriptstyleoldsymbol{u}(oldsymbol{x})=(u_1(oldsymbol{x}),ldots,u_p(oldsymbol{x})) be a function in BV(Omega) with Omega being an open subset of scriptstylemathbb{R}^n .Then scriptstyle fcircoldsymbol{u}(oldsymbol{x})=f(oldsymbol{u}(oldsymbol{x}))in BV(Omega) and

:frac{partial f(oldsymbol{u}(oldsymbol{x}))}{partial x_i}=sum_{k=1}^pfrac{partialar{f}(oldsymbol{u}(oldsymbol{x}))}{partial u_k}frac{partial{u_k(oldsymbol{x}){partial x_i}qquadforall i=1,ldots,n

where scriptstylear f(oldsymbol{u}(oldsymbol{x})) is the mean value of the function at the point scriptstyle x inOmega, defined as

:ar f(oldsymbol{u}(oldsymbol{x}))=int_0^1 fleft(oldsymbol{u}_{oldsymbol{hat a(oldsymbol{x})t + oldsymbol{u}_{-oldsymbol{hat a(oldsymbol{x})(1-t) ight)dt

A more general chain rule formula for Lipschitz continuous functions scriptstyle f:mathbb{R}^p ightarrowmathbb{R}^s has been found by Luigi Ambrosio and Gianni Dal Maso and published in the paper Harv|Ambrosio|Dal Maso|1990. However, even this formula has very important direct consequences: choosing scriptstyle f(u)=v(oldsymbol{x})u(oldsymbol{x}) where scriptstyle v(oldsymbol{x}) is a "BV" function the preceding formula becomes the "Leibnitz rule" for BV functions

:frac{partial v(oldsymbol{x})u(oldsymbol{x})}{partial x_i} = {ar u(oldsymbol{x})}frac{partial v(oldsymbol{x})}{partial x_i} + {ar v(oldsymbol{x})}frac{partial u(oldsymbol{x})}{partial x_i}

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

BV(&Omega;) is a Banach algebra

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

::egin{matrix} vu_nxrightarrow [n oinfty] {} vu \ v_nuxrightarrow [n oinfty] {} vu end{matrix}qquad vuin BV(Omega)

therefore the ordinary product of functions is continuous in BV(Omega) respect to each argument, making this function space a Banach algebra.

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 scriptstyle varphi : [0, +infty)longrightarrow [0, +infty) be any increasing function such that scriptstyle varphi(0) = varphi(0+) =lim_{x ightarrow 0_+}varphi(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 scriptstyle oldsymbolvarphi-variation of f over [0, T] is defined as

:mathop{varphimbox{-Var_{ [0, T] } (f) := sup sum_{j = 0}^{k} varphi left( | f(t_{j + 1}) - f(t_{j}) |_{X} 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

:0 = t_{0} < t_{1} < ldots < t_{k} = T.

The original notion of variation considered above is the special case of scriptstyle varphi-variation for which the weight function is the identity function: therefore an integrable function f is said to be a weighted "BV" function (of weight scriptstylevarphi) if and only if its scriptstyle varphi-variation is finite.

:fin BV_varphi( [0, T] ;X)iff mathop{varphimbox{-Var_{ [0, T] } (f) <+infty

The space scriptstyle BV_varphi( [0, T] ;X) is a topological vector space with respect to the norm

:| f |_{BV_varphi} := | f |_{infty} + mathop{varphi mbox{-Var_{ [0, T] } (f),

where scriptstyle| f |_{infty} denotes the usual supremum norm of "f". Weighted "BV" functions were introduced and studied in full generality by Wladislav Orlicz and Julian Musielak in the paper Harv|Musielak|Orlicz|1959: Laurence Chisholm Young studied earlier the case scriptstylevarphi(x)=x^p where "p" is a positive integer.

"SBV" functions

SBV functions "i.e." "Special functions of Bounded Variation" where introduced by Luigi Ambrosio and Ennio de Giorgi in the paper Harv|Ambrosio|De Giorgi|1988, dealing with free discontinuity variational problems: given an open subset Omega of scriptstylemathbb{R}^n , the space SBV(Omega) is a proper subspace of 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 locally integrable function u, then scriptstyle uin {S!BV}(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

: int_Omegavert fvert dH^n+ int_Omegavert gvert dH^{n-1}<+infty.

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 H^alpha is the alpha-dimensional Hausdorff measure.

Details on the properties of "SBV" functions can be found in works cited in the bibliography section: particularly the paper Harv|De Giorgi|1992 contains a useful bibliography.

"bv" sequences

As particular examples of Banach spaces, harvtxt|Dunford|Schwartz|1958|loc=Chapter IV consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence "x"=("x"i) of real or complex numbers is defined by: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:|x|_{bv} = |x_1| + TV(x) = |x_1| + sum_{i=1}^infty |x_{i+1}-x_i|.With this norm, the space "bv" is a Banach space.

The total variation itself defines a norm on a certain subspace of "bv", denoted by "bv"0, consisting of sequences "x" = ("x"i) for which:lim_{n oinfty} x_n =0.The norm on "bv"0 is denoted:|x|_{bv_0} = TV(x) = sum_{i=1}^infty |x_{i+1}-x_i|.With respect to this norm "bv"0 becomes a Banach space as well.

Examples

The function

:f(x) = egin{cases} 0, & mbox{if }x =0 \ sin(1/x), & mbox{if } x eq 0 end{cases}

is "not" of bounded variation on the interval [0, 2/pi]

While it is harder to see, the function

:f(x) = egin{cases} 0, & mbox{if }x =0 \ x sin(1/x), & mbox{if } x eq 0 end{cases}

is "not" of bounded variation on the interval [0, 2/pi] either.

At the same time, the function

:f(x) = egin{cases} 0, & mbox{if }x =0 \ x^2 sin(1/x), & mbox{if } x eq 0 end{cases}

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

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 forall 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}.

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'(x) exists almost everywhere (i.e. except for a set of measure zero).
* Minimal surfaces turn out very often to be graphs of "BV" functions: in this context, see reference Harv|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 Harv|Hudjaev|Vol'pert|1986 details a very ample set of mathematical physics applications of "BV" functions. Also there is some modern application which deserves a brief description.

*The Mumford-Shah functional: the segmentation problem for a two-dimensional image, i.e. the problem of faithful reproduction of contours and grey scales is equivalent to the minimization of such functional.

See also

* Total variation
* Caccioppoli set
* "L""p"(&Omega;) space
* Lebesgue-Stieltjes integral, Riemann-Stieltjes integral
* Radon measure
* Reduced derivative
* Helly's selection theorem
* Renato Caccioppoli, Lamberto Cesari, Ennio de Giorgi

References

*Harvrefcol
Surname = Cesari
Given = Lamberto
Year = 1986
Chapter = 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)
Editor = G. Montalenti et als.
Title = 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)
Publisher = Accademia Nazionale dei Lincei, [http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32847 Atti dei Convegni Lincei, Vol. 77]
URL = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847
Place = Rome, 6-9 June 1985
Pages = 41-73
. Some recollections from one of the founders of the theory of "BV" functions of several variables (in Italian).
*Harvrefcol
Surname1 = Dunford
Given1 = Nelson
Surname2 = Schwartz
Given2 = Jacob T.
Title = Linear operators. Part I: General Theory
Publisher = Wiley-Interscience
Place = New York-London-Sydney
Year = 1958
. Includes a discussion of the functional-analytic properties of spaces of functions of bounded variation.
*Harvrefcol
Surname = Giusti
Given = Enrico
Title = Minimal surfaces and functions of bounded variations
Publisher = [http://www.birkhauser.com Birkhäuser Verlag]
Place = Basel
Year = 1984
ISBN 0-8176-3153-4, particularly part I, chapter 1 "Functions of bounded variation and Caccioppoli sets".
*Harvrefcol
Surname1 = Hudjaev
Given1 = Sergei Ivanovich
Surname2 = Vol'pert
Given2 = Aizik Isaakovich
Title = Analysis in classes of discontinuous functions and equations of mathematical physics
Publisher = Martinus Nijhoff Publishers
Place = Dordrecht
Year=1986
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.
*Harvrefcol
Surname1 = Kannan
Given1 = Rangachary
Surname2 = King Krueger
Given2 = Carole
Title = Advanced analysis on the real line
Publisher = Springer Verlag
Place = Berlin-Heidelberg-New York
Year = 1996
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.
*citation
first=Andrej N.
last=Kolmogorov
first2=Sergej V.
last2=Fomin
title=Introductory Real Analysis
publisher=Dover Publications
place=New York
year=1969
id=ISBN 0486612260

*Harvrefcol
Surname1 = Màlek
Given1 = Josef
Surname2 = Nečas
Given2 = Jindřich
Surname3 = Rokyta
Given3 = Mirko
Surname4 = Růžička
Given4 = Michael
Title = Weak and measure-valued solutions to evolutionary PDEs
Publisher = Chapman & Hall/CRC Press
Place = London-Weinheim-New York-Tokyo-Melbourne-Madras
Year = 1996
ISBN 0-412-57750-X. One of the most complete monographs on the theory of Young measures, strongly oriented to applications in continuum mechanics of fluids.
* Harvrefcol
Surname = Moreau
Given = Jean Jacques
Year= 1988
Chapter = Bounded variation in time.
Editor= Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G.
Title = Topics in nonsmooth mechanics.
Pages = 1-74
Publisher = [http://www.birkhauser.com Birkhäuser Verlag]
Place = Basel
ISBN 3-7643-1907-0.
*Harvrefcol
Surname1 = Musielak
Given1 = Julian
Surname2 = Orlicz
Given2 = Wladislaw
Title = [http://matwbn.icm.edu.pl/ksiazki/sm/sm18/sm1812.pdf On generalized variations (I)]
Journal = [http://matwbn.icm.edu.pl/spis.php?wyd=2&jez= Studia Mathematica]
Volume = 18
Page = 13-41
Place = Warszawa-Wrocław
Year = 1959
. The first paper where weighted "BV" functions are studied in full generality.
*citation
first=Frigyes
last=Riesz
first2=Béla
last2=Szőkefalvi-Nagy
title=Functional Analysis
publisher=Dover Publications
place=New York
year=1990
id=ISBN 0-486-66289-6

*Harvrefcol
Surname = Vol'pert
Given = Aizik Isaakovich
Title = [http://www.math.technion.ac.il/~volp/spaces_BV.pdf Spaces BV and quasi-linear equations]
Journal = [http://www.turpion.org/php/homes/pa.phtml?jrnid=sm Mathematics USSR-Sbornik]
Volume = 2
Issue = 2
Page = 225-267
Year = 1967
Access-date = January 23, 2007
. A seminal paper where Caccioppoli sets and "BV" functions are deeply studied and applied to the theory of partial differential equations.

Bibliography

*Harvrefcol
Surname1 = Alberti
Given1 = Giovanni
Surname2 = Mantegazza
Given2 = Carlo
Title = A note on the theory of SBV functions
Journal = [http://umi.dm.unibo.it/italiano/Editoria/BollettinoB.html Bollettino Unione Matematica Italiana, Sezione B]
Volume = 7
Year = 1997
Page = 375-382
. A paper containing a demonstration of the compactness of the set of SBV functions.
*Harvrefcol
Surname1 = Ambrosio
Given1 = Luigi
Surname2 = Dal Maso
Given2 = Giovanni
Title = [http://links.jstor.org/sici?sici=0002-9939%28199003%29108%3A3%3C691%3AAGCRFD%3E2.0.CO%3B2-3&size=LARGE&origin=JSTOR-enlargePage A General Chain Rule for Distributional Derivatives]
Journal = [http://www.ams.org/journals/proc/ Proceedings of the American Mathematical Society]
Volume = 108
Issue = 3
Year = 1990
Page = 691-702
. DOI 10.2307/2047789. A paper containing a very general chain rule formula for composition of BV functions.
*Harvrefcol
Surname1 = Ambrosio
Given1 = Luigi
Surname2 = De Giorgi
Given2 = Ennio
Title = Un nuovo tipo di funzionale del calcolo delle variazioni (A new kind of functional in the calculus of variations)
Journal = Atti dell'Accademia Nazionale dei Lincei, [http://www.lincei.it/pubblicazioni/rendicontiFMN/cliccami_eng.htm Rendiconti Lincei, Classe di Scienze Fisiche, Mathematiche, Naturali]
Volume = 82
Year = 1988
Page = 199-210
(in Italian). The first paper about "SBV" functions and related variational problems.
*Harvrefcol
Surname1 = Conway
Given1 = Edward D
Surname2 = Smoller
Given2 = Joel A.
Title = Global solutions of the Cauchy problem for quasi-linear first-order equations in several space variables
Journal = [http://www3.interscience.wiley.com/cgi-bin/jhome/29240 Communications on Pure and Applied Mathematics]
Volume = 19
Year = 1966
Page = 95-105
. An important paper where properties of "BV" functions were applied to "single" hyperbolic equations of first order.
*Harvrefcol
Surname = De Giorgi
Given = Ennio
Year = 1992
Chapter = Problemi variazionali con discontinuità libere (Free-discontinuity variational problems)
Editor = E. Amaldi et als.
Title = Convegno internazionale in memoria di Vito Volterra (International congress in memory of Vito Volterra)
Publisher = Accademia Nazionale dei Lincei, [http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862 Atti dei Convegni Lincei, Vol. 92]
Place = Roma, 8-11 October 1990
Pages = 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.

*Tony F. Chan and Jackie (Jianhong) Shen (2005), [http://jackieneoshen.googlepages.com/ImagingNewEra.html "Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods"] , SIAM Publisher, ISBN 089871589X (with in-depth coverage and extensive applications of Bounded Variations in modern image processing, as started by Rudin, Osher, and Fatemi).

External links

Theory

* Boris I. Golubov (and comments of Anatolii Georgievich Vitushkin) " [http://eom.springer.de/V/v096110.htm Variation of a function] ", Springer-Verlag Online Encyclopaedia of Mathematics.
*.
*Harvrefcol
Surname = Jordan
Given = Camille
Title = [http://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227.chemindefer Sur la série de Fourier]
Journal = Comptes rendus des Académie des sciences de Paris
Volume = 92
Place = Paris, janv.-juin
Year = 1881
Page = 228-230
Access-date=January 23, 2007
(at Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.
* Rowland, Todd and Weisstein, Eric W. " [http://mathworld.wolfram.com/BoundedVariation.html Bounded Variation] ". From MathWorld&mdash;A Wolfram Web Resource.

Other

* Luigi Ambrosio [http://cvgmt.sns.it/people/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.
* [http://cvgmt.sns.it/ Research Group in Calculus of Variations and Geometric Measure Theory] , Scuola Normale Superiore, Pisa.
* [http://www.math.technion.ac.il/people/volp/index.html Aizik Isaakovich Vol'pert] at Technion. Academic home page of one of the leading contributors to the theory of "BV" functions.----


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