Editing Bounded variation (section)

===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]].