Editing Bounded variation (section)

==History==
According to Boris 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 successful step in the generalization of this concept to functions of several variables was due to [[Leonida Tonelli]],<ref>[[Leonida Tonelli|Tonelli]] introduced what is now called after him '''Tonelli plane variation''': for an analysis of this concept and its relations to other generalizations, see the entry "[[Total variation]]".</ref> who introduced a class of ''continuous'' BV functions in 1926 {{Harv|Cesari|1986|pp=47–48}}, to extend his [[Direct method in the calculus of variations|direct method]] for finding solutions to problems in the [[calculus of variations]] in more than one variable. Ten years after, in {{Harv|Cesari|1936}}, [[Lamberto Cesari]] ''changed the continuity requirement'' in Tonelli's definition ''to a less restrictive [[integral|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 theory|measure]] of [[smooth function|nonsmooth]] [[boundary (topology)|boundaries]] of [[set (mathematics)|sets]] (see the entry "''[[Caccioppoli set]]''" for further information). [[Olga Arsenievna Oleinik]] introduced her view of generalized solutions for [[nonlinear partial differential equation]]s as functions from the space BV in the paper {{Harv|Oleinik|1957}}, and was able to construct a generalized solution of bounded variation of a [[First-order partial differential equation|first order]] partial differential equation in the paper {{Harv|Oleinik|1959}}: few years later, [[Edward D. Conway]] and [[Joel A. Smoller]] applied BV-functions to the study of a single [[hyperbolic equation|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 [[Cauchy boundary condition|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}} he proved the [[Bounded variation#Chain rule for BV functions|chain rule for BV functions]] and in the book {{Harv|Hudjaev|Vol'pert|1985}} he, jointly with his pupil [[Sergei Ivanovich Hudjaev]], explored extensively the properties of BV functions and their application. His chain rule formula was later extended by [[Luigi Ambrosio]] and [[Gianni Dal Maso]] in the paper {{Harv|Ambrosio|Dal Maso|1990}}.