Editing Bounded variation (section)

==Formal definition==

=== BV functions of one variable ===
{{EquationRef|1|Definition 1.1.}} The '''[[total variation]]''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] ''f'', defined on an [[interval (mathematics)|interval]] <math>[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 display="inline"> \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\} </math> of all [[partition of an interval|partitions]] of the interval considered.

If ''f'' is [[derivative|differentiable]] and its derivative is Riemann-integrable, its total variation is the vertical component of the [[arc length|arc-length]] of its graph, that is to say,

:<math> V_a^b(f) = \int _a^b |f'(x)|\,\mathrm{d}x.</math>

{{EquationRef|2|Definition 1.2.}} A  real-valued function <math> f </math> on the [[real line]] is said to be of '''bounded variation''' ('''BV function''') on a chosen [[interval (mathematics)|interval]] <math>[a,b] \subset \mathbb{R}</math>  if its total variation is finite, ''i.e.''
:<math> f \in \operatorname{BV}([a,b]) \iff V_a^b(f) < +\infty </math>

It can be proved that a real function <math>f</math> is of bounded variation in <math>[a,b]</math> if and only if it can be written as the difference <math>f=f_1-f_2</math> of two non-decreasing functions <math>f_1</math> and <math>f_2</math> on <math>[a,b]</math>: this result is known as the [https://www.encyclopediaofmath.org/index.php/Jordan_decomposition_(of_a_function) Jordan decomposition of a function] and it is related to the [[Hahn decomposition theorem#Jordan measure decomposition|Jordan decomposition of a measure]].

Through the [[Stieltjes integral]], any function of bounded variation on a closed interval <math>[a, b]</math> defines a [[bounded linear functional]] on <math>C([a, b])</math>. In this special case,<ref>See for example {{harvtxt|Kolmogorov|Fomin|1969|pp=374–376}}.</ref> the [[Riesz–Markov–Kakutani representation theorem]] states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or [[probability measure]]s correspond to positive non-decreasing lower [[semicontinuous function]]s. This point of view has been important in
[[spectral theory]],<ref>For a general reference on this topic, see {{harvtxt|Riesz|Szőkefalvi-Nagy|1990}}</ref> in particular in its application to [[spectral theory of ordinary differential equations|ordinary differential equations]].

===BV functions of several variables===
Functions of bounded variation, BV [[function (mathematics)|functions]], are functions whose distributional [[directional derivative|derivative]] is a [[Wikt:finite|finite]]<ref>In this context, "finite" means that its value is never [[Infinity|infinite]], i.e. it is a [[finite measure]].</ref> [[Radon measure]]. More precisely:

{{EquationRef|3|Definition 2.1.}} Let '''<math> \Omega </math>''' be an [[open subset]] of <math>\mathbb{R}^n</math>. A function '''<math> u </math>''' belonging to '''[[Lp space|<math>L^1(\Omega)</math>]]''' is said to be of '''bounded variation''' ('''BV function'''), and written

:<math> u\in \operatorname\operatorname{BV}(\Omega)</math>

if there exists a [[Finite measure|finite]] [[vector-valued function|vector]] [[Radon measure]] <math>  Du\in\mathcal M(\Omega,\mathbb{R}^n)</math> such that the following equality holds

:<math>
\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)
</math>

that is, '''<math>u</math>''' defines a [[linear functional]] on the space <math>  C_c^1(\Omega,\mathbb{R}^n)</math> of [[Smooth function|continuously differentiable]] [[Vector-valued function|vector functions]] <math> \boldsymbol{\phi} </math> of [[support (mathematics)#Compact support|compact support]] contained in '''<math> \Omega </math>''': the vector [[measure (mathematics)|measure]] '''<math>Du</math>''' represents therefore the [[Distribution (mathematics)#Definitions of test functions and distributions|distributional]] or [[weak derivative|weak]] [[gradient]] of '''<math>u</math>'''.

BV can be defined equivalently in the following way.

{{EquationRef|4|Definition 2.2.}} Given a function '''<math>u</math>''' belonging to '''<math>L^1(\Omega)</math>''', the '''total variation of <math>u</math>'''<ref name="Tvar">See the entry "[[Total variation]]" for further details and more information.</ref> in <math>\Omega</math> is defined as

:<math> 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)}\le 1\right\}</math>

where <math>  \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]]. Sometimes, especially in the theory of [[Caccioppoli set]]s, the following notation is used

:<math>\int_\Omega\vert D u\vert = V(u,\Omega)</math>

in order to emphasize that <math>V(u,\Omega)</math> is the total variation of the [[Distribution (mathematics)#Definitions of test functions and distributions|distributional]] / [[weak derivative|weak]] [[gradient]] of '''<math>u</math>'''. This notation reminds also that if '''<math>u</math>''' is of class  '''<math>C^1</math>''' (i.e. a [[continuous function|continuous]] and [[differentiable function]] having [[continuous function|continuous]] [[derivative]]s) then its [[Total variation|variation]] is exactly the [[Integral (measure theory)|integral]] of the [[absolute value]] of its [[gradient]].

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

:<math> \operatorname\operatorname{BV}(\Omega)=\{ u\in L^1(\Omega)\colon V(u,\Omega)<+\infty\}</math>

The two definitions are equivalent since if <math>V(u,\Omega)<+\infty </math> then

:<math>\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)
</math>

therefore <math display="inline"> \displaystyle \boldsymbol{\phi}\mapsto\,\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, dx</math> defines a [[continuous linear functional]] on the space <math>C_c^1(\Omega,\mathbb{R}^n)</math>. Since <math>C_c^1(\Omega,\mathbb{R}^n)
\subset C^0(\Omega,\mathbb{R}^n)</math> as a [[linear subspace]], this [[continuous linear functional]] can be extended [[continuous function|continuously]] and [[linearity|linearly]] to the whole <math>C^0(\Omega,\mathbb{R}^n)</math> by the [[Hahn–Banach theorem]]. Hence the continuous linear functional defines a [[Radon measure#Duality|Radon measure]] by the [[Riesz–Markov–Kakutani representation theorem]].

===Locally BV functions===
If the [[function space]] of [[locally integrable function]]s, i.e. [[Function (mathematics)|function]]s belonging to <math>  L^1_\text{loc}(\Omega)</math>, is considered in the preceding definitions {{EquationNote|2|1.2}}, {{EquationNote|3|2.1}} and {{EquationNote|4|2.2}} instead of the one of [[integrable function|globally integrable functions]], then the function space defined is that of '''functions of locally bounded variation'''. Precisely, developing this idea for {{EquationNote|4|definition 2.2}}, a '''[[local property|local]] variation''' is defined as follows,

: <math> 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)}\le 1\right\}</math>

for every [[Set (mathematics)|set]] <math>  U\in\mathcal{O}_c(\Omega)</math>, having defined <math>  \mathcal{O}_c(\Omega)</math> as the set of all [[Relatively compact subspace|precompact]] [[open subset]]s of '''<math>\Omega</math>''' with respect to the standard [[topology]] of [[dimension (mathematics)|finite-dimensional]] [[vector space]]s, and correspondingly the class of functions of locally bounded variation is defined as
:<math>\operatorname{BV}_\text{loc}(\Omega)=\{ u\in L^1_\text{loc}(\Omega)\colon \, (\forall U\in\mathcal{O}_c(\Omega)) \, V(u,U)<+\infty\}</math>

===Notation===
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 {{Harvtxt|Giusti|1984}} (partially), {{Harvtxt|Hudjaev|Vol'pert|1985}} (partially), {{Harvtxt|Giaquinta|Modica|Souček|1998}} and is the following one
*<math> \operatorname\operatorname{BV}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of globally bounded variation
*<math> \operatorname\operatorname{BV}_{\text{loc}}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of locally bounded variation
The second one, which is adopted in references {{Harvtxt|Vol'pert|1967}} and {{Harvtxt|Maz'ya|1985}} (partially), is the following:
*<math> \overline{\operatorname\operatorname{BV}}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of globally bounded variation
*<math> \operatorname\operatorname{BV}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of locally bounded variation