Editing Bounded variation (section)

==Basic properties==
Only the properties common to [[Function (mathematics)|function]]s of one variable and to [[Function (mathematics)|function]]s of several variables will be considered in the following, and [[Mathematical proof|proof]]s will be carried on only for functions of several variables since the [[Mathematical proof|proof]] for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References {{Harv|Giusti|1984|pp=7–9}}, {{Harv|Hudjaev|Vol'pert|1985}} and {{Harv|Màlek|Nečas|Rokyta|Růžička|1996}} are extensively used.

===BV functions have only jump-type or removable discontinuities===
In the case of one variable, the assertion is clear: for each point <math>x_0</math> in the [[interval (mathematics)|interval]] <math>[a , b]\subset\mathbb{R}</math> of definition of the function '''<math>u</math>''', either one of the following two assertions is true

:<math> \lim_{x\rightarrow x_{0^-}}\!\!\!u(x) = \!\!\!\lim_{x\rightarrow x_{0^+}}\!\!\!u(x) </math>
:<math> \lim_{x\rightarrow x_{0^-}}\!\!\!u(x) \neq \!\!\!\lim_{x\rightarrow x_{0^+}}\!\!\!u(x) </math>

while both [[Limit of a function|limits]] exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a [[Linear continuum|continuum]] of [[Direction (geometry, geography)|direction]]s along which it is possible to approach a given point '''<math>x_0</math>''' belonging to the domain '''<math>\Omega</math>'''⊂<math>\mathbb{R}^n</math>. It is necessary to make precise a suitable concept of [[Limit of a function|limit]]: choosing a [[unit vector]] <math>{\boldsymbol{\hat{a}}}\in\mathbb{R}^n</math> it is possible to divide '''<math>\Omega</math>''' in two sets

:<math>\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)} = \Omega \cap \{\boldsymbol{x}\in\mathbb{R}^n|\langle\boldsymbol{x}-\boldsymbol{x}_0,{\boldsymbol{\hat{a}}}\rangle>0\} \qquad
\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)} = \Omega \cap \{\boldsymbol{x}\in\mathbb{R}^n|\langle\boldsymbol{x}-\boldsymbol{x}_0,-{\boldsymbol{\hat{a}}}\rangle>0\} </math>

Then for each point '''<math>x_0</math>''' belonging to the domain <math>\Omega\in\mathbb{R}^n</math> of the BV function '''<math>u</math>''', only one of the following two assertions is true

:<math> \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x}) = \!\!\!\!\!\!\!\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})
</math>
:<math>  \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x}) \neq \!\!\!\!\!\!\!\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})
</math>

or '''<math>x_0</math>''' belongs to a [[subset]] of '''<math>\Omega</math>''' having zero <math>n-1</math>-dimensional [[Hausdorff measure]]. The quantities

:<math>\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x})=u_{\boldsymbol{\hat a}}(\boldsymbol{x}_0) \qquad \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})=u_{-\boldsymbol{\hat a}}(\boldsymbol{x}_0)</math>

are called '''approximate limits''' of the BV function '''<math>u</math>''' at the point '''<math>x_0</math>'''.

===''V''(&sdot;,&nbsp;&Omega;) is lower semi-continuous on ''L''<sup>1</sup>(&Omega;)===
The [[functional (mathematics)|functional]] <math>V(\cdot,\Omega):\operatorname\operatorname{BV}(\Omega)\rightarrow \mathbb{R}^+</math> is [[semi-continuity|lower semi-continuous]]:
to see this, choose a [[Cauchy sequence]] of BV-functions '''<math>\{u_n\}_{n\in\mathbb{N}}</math>''' converging to '''[[locally integrable function|<math>u\in L^1_\text{loc}(\Omega)</math>]]'''. Then, since all the functions of the sequence and their limit function are [[integral|integrable]] and by the definition of [[lower limit]]

:<math>\begin{align}
\liminf_{n\rightarrow\infty}V(u_n,\Omega) 
&\geq \liminf_{n\rightarrow\infty} \int_\Omega u_n(x)\operatorname{div}\, \boldsymbol{\phi}\, \mathrm{d}x \\
&\geq \int_\Omega \lim_{n\rightarrow\infty} u_n(x)\operatorname{div}\, \boldsymbol{\phi}\, \mathrm{d}x \\
&= \int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}\, \mathrm{d}x \qquad\forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n),\quad\Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\leq 1 
\end{align}</math>

Now considering the [[supremum]] on the set of functions <math>\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)</math> such that <math>\Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\leq 1 </math> then the following inequality holds true

:<math>\liminf_{n\rightarrow\infty}V(u_n,\Omega)\geq V(u,\Omega)</math>

which is exactly the definition of [[semicontinuity|lower semicontinuity]].

===BV(&Omega;) is a Banach space===
By definition '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is a [[subset]] of '''[[integrable function|<math>L^1(\Omega)</math>]]''', while [[linearity]] follows from the linearity properties of the defining [[integral]] i.e.

:<math>\begin{align}
\int_\Omega [u(x)+v(x)]\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x & =
\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x +\int_\Omega v(x) \operatorname{div} \boldsymbol{\phi}(x)\,\mathrm{d}x = \\
& =- \int_\Omega \langle\boldsymbol{\phi}(x), Du(x)\rangle- \int_\Omega \langle \boldsymbol{\phi}(x), Dv(x)\rangle =- \int_\Omega \langle \boldsymbol{\phi}(x), [Du(x)+Dv(x)]\rangle
\end{align}
</math>

for all <math>\phi\in C_c^1(\Omega,\mathbb{R}^n)</math> therefore <math>u+v\in \operatorname\operatorname{BV}(\Omega)</math>for all <math>u,v\in \operatorname\operatorname{BV}(\Omega)</math>, and

:<math>
\int_\Omega c\cdot u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x =
c \int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x =
-c \int_\Omega \langle \boldsymbol{\phi}(x), Du(x)\rangle
</math>

for all <math> c\in\mathbb{R}</math>, therefore <math>  cu\in \operatorname\operatorname{BV}(\Omega)</math> for all <math>  u\in \operatorname\operatorname{BV}(\Omega)</math>, and all <math> c\in\mathbb{R}</math>. The proved [[vector space]] properties imply that '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is a [[vector subspace]] of '''[[Lp space|<math>L^1(\Omega)</math>]]'''. Consider now the function <math>\|\;\|_{\operatorname{BV}}:\operatorname\operatorname{BV}(\Omega)\rightarrow\mathbb{R}^+</math> defined as

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

where <math>\| \; \|_{L^1}</math> is the usual '''[[Lp space#Lp spaces and Lebesgue integrals|<math>L^1(\Omega)</math> norm]]''': it is easy to prove that this is a [[norm (mathematics)|norm]] on '''<math>\operatorname\operatorname{BV}(\Omega)</math>'''. To see that '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is [[complete metric space|complete]] respect to it, i.e. it is a [[Banach space]], consider a [[Cauchy sequence]] <math>\{u_n\}_{n\in\mathbb{N}}</math> in '''<math>\operatorname\operatorname{BV}(\Omega)</math>'''. By definition it is also a [[Cauchy sequence]] in '''<math>L^1(\Omega)</math>''' and therefore has a [[limit of a sequence|limit]] '''<math>u</math>''' in '''<math>L^1(\Omega)</math>''': since '''<math>u_n</math>''' is bounded in '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' for each '''<math>n</math>''', then <math>\Vert u \Vert_{\operatorname{BV}} < +\infty </math> by [[semicontinuity|lower semicontinuity]] of the variation <math>V(\cdot,\Omega)</math>, therefore '''<math>u</math>''' is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number '''<math>\varepsilon</math>'''

:<math>\Vert u_j - u_k \Vert_{\operatorname{BV}}<\varepsilon\quad\forall j,k\geq N\in\mathbb{N} \quad\Rightarrow\quad V(u_k-u,\Omega)\leq \liminf_{j\rightarrow +\infty} V(u_k-u_j,\Omega)\leq\varepsilon</math>
From this we deduce that <math>V(\cdot,\Omega)</math> is continuous because it's a norm.

===BV(&Omega;) is not separable===
To see this, it is sufficient to consider the following example belonging to the space '''<math>\operatorname\operatorname{BV}([0,1])</math>''':<ref>The example is taken from  {{Harvtxt|Giaquinta|Modica|Souček|1998|p=331}}: see also {{harv|Kannan|Krueger|1996|loc=example 9.4.1, p. 237}}.</ref> for each 0&nbsp;<&nbsp;''&alpha;''&nbsp;<&nbsp;1 define
:<math>\chi_\alpha=\chi_{[\alpha,1]}=
\begin{cases} 0 & \mbox{if } x \notin\; [\alpha,1] \\ 
              1 & \mbox{if } x \in [\alpha,1]
\end{cases}
</math>
as the [[indicator function|characteristic function]] of the [[Interval (mathematics)#Definitions|left-closed interval]] <math>[\alpha,1]</math>. Then, choosing <math>\alpha,\beta \in [0,1]</math> such that <math>\alpha \ne \beta</math> the following relation holds true:
:<math>\Vert \chi_\alpha - \chi_\beta \Vert_{\operatorname{BV}}=2</math>
Now, in order to prove that every [[Dense set|dense subset]] of '''<math>\operatorname\operatorname{BV}(]0,1[)</math>''' cannot be [[countable set|countable]], it is sufficient to see that for every <math>\alpha\in[0,1]</math> it is possible to construct the [[Ball (mathematics)|ball]]s
:<math>B_\alpha=\left\{\psi\in \operatorname\operatorname{BV}([0,1]);\Vert \chi_\alpha - \psi \Vert_{\operatorname{BV}}\leq 1\right\}</math>
Obviously those balls are [[Disjoint sets|pairwise disjoint]], and also are an [[indexed family]] of [[set (mathematics)|set]]s whose [[index set]] is <math>[0,1]</math>. This implies that this family has the [[cardinality of the continuum]]: now, since every dense subset of <math>\operatorname\operatorname{BV}([0,1])</math> must have at least a point inside each member of this family, its cardinality is at least that of the continuum and therefore cannot a be countable subset.<ref>The same argument is used by {{Harvtxt|Kolmogorov|Fomin|1969|loc=example 7, pp. 48–49}}, in order to prove the non [[Separable space|separability]] of the space of [[bounded sequence]]s, and also {{harvtxt|Kannan|Krueger|1996|loc=example 9.4.1, p. 237}}.</ref> This example can be obviously extended to higher dimensions, and since it involves only [[Local property|local properties]], it implies that the same property is true also for '''<math>\operatorname{BV}_{loc}</math>'''.

===Chain rule for locally BV(&Omega;) functions===
[[Chain rule]]s for [[smooth function|nonsmooth function]]s are very important in [[mathematics]] and [[mathematical physics]] since there are several important [[Mathematical model|physical model]]s whose behaviors are described by [[Function (mathematics)|functions]] or [[functional (mathematics)|functional]]s with a very limited degree of [[Smooth function|smoothness]]. The following chain rule is proved in the paper {{Harv|Vol'pert|1967|p=248}}. Note all [[partial derivative]]s must be interpreted in a generalized sense, i.e., as [[Generalized derivative#Basic idea|generalized derivative]]s.

'''Theorem'''. Let <math>f:\mathbb{R}^p\rightarrow\mathbb{R}</math> be a function of class '''<math>C^1</math>''' (i.e. a [[continuous function|continuous]] and [[differentiable function]] having [[continuous function|continuous]] [[derivative]]s) and let <math>\boldsymbol{u}(\boldsymbol{x})=(u_1(\boldsymbol{x}),\ldots,u_p(\boldsymbol{x})) </math> be a function in '''<math>\operatorname\operatorname{BV}_{loc} (\Omega)</math>''' with '''<math> \Omega </math>''' being an [[open subset]] of <math> \mathbb{R}^n </math>.
Then <math>f\circ\boldsymbol{u}(\boldsymbol{x})=f(\boldsymbol{u}(\boldsymbol{x}))\in \operatorname\operatorname{BV}_{loc} (\Omega) </math> and

:<math>\frac{\partial f(\boldsymbol{u}(\boldsymbol{x}))}{\partial x_i}=\sum_{k=1}^p\frac{\partial\bar{f}(\boldsymbol{u}(\boldsymbol{x}))}{\partial u_k}\frac{\partial{u_k(\boldsymbol{x})}}{\partial x_i}
\qquad\forall i=1,\ldots,n</math>

where <math>\bar f(\boldsymbol{u}(\boldsymbol{x}))</math> is the mean value of the function at the point '''<math>x \in\Omega</math>''', defined as

:<math>\bar f(\boldsymbol{u}(\boldsymbol{x})) = \int_0^1 f\left(\boldsymbol{u}_{\boldsymbol{\hat a}}(\boldsymbol{x})t + \boldsymbol{u}_{-\boldsymbol{\hat a}}(\boldsymbol{x})(1-t)\right) \, dt</math>

A more general [[chain rule]] [[formula]] for [[lipschitz continuity|Lipschitz continuous functions]] <math>f:\mathbb{R}^p\rightarrow\mathbb{R}^s</math> has been found by [[Luigi Ambrosio]] and [[Gianni Dal Maso]] and is published in the paper {{Harv|Ambrosio|Dal Maso|1990}}. However, even this formula has very important direct consequences: we use <math>( u(\boldsymbol{x}), v(\boldsymbol{x}))</math> in place of <math>\boldsymbol u(\boldsymbol{x})</math>, where <math>v(\boldsymbol{x})</math> is also a <math>BV_{loc}</math> function. We have to assume also that <math>\bar u(\boldsymbol{x})</math> is locally integrable with respect to the measure <math>\frac{\partial v(\boldsymbol{x})}{\partial x_i}</math> for each <math>i</math>, and that <math>\bar v(\boldsymbol{x})</math> is locally integrable with respect to the measure <math>\frac{\partial u(\boldsymbol{x})}{\partial x_i}</math> for each <math>i</math>. Then choosing <math>f((u,v))=uv</math>, the preceding formula gives the '''''[[Product rule|Leibniz rule]]''''' for 'BV' functions

:<math>\frac{\partial v(\boldsymbol{x})u(\boldsymbol{x})}{\partial x_i} = {\bar u(\boldsymbol{x})}\frac{\partial v(\boldsymbol{x})}{\partial x_i} +
{\bar v(\boldsymbol{x})}\frac{\partial u(\boldsymbol{x})}{\partial x_i} </math>