Bounded Variation Functions

Functions with bounded variations play an important role in recent modern mathematical analysis (especially since the end of the 20th century). Spaces of functions with bounded variations are positioned "finely" between spaces of Sobolev functions that are too restricted (not enough functions, notably not "jump" type functions) and spaces of Lebesgue functions that are too vast (too many functions).

They are frequently used to define generalized solutions to nonlinear minimization problems involving functionals, ordinary differential equations, partial differential equations and integral equations, in mathematics, physics and engineering (mechanics, image processing, etc.).

It's important to note that real-world applications require increasingly "sophisticated" theoretical mathematical notions and tools, in order to obtain increasingly precise solutions to engineering problems.