Overview
ABSTRACT
Functions with bounded variations are particular integrable functions whose total variations are finite. They play an important role in modern mathematical analysis. Functions with bounded variations of a single variable and with several variables are presented, with examples and counterexamples. Functions with bounded local variations are introduced succinctly. Another part is devoted to finite distributional perimeter sets (i.e. Caccioppoli sets), as well as the presentation of generalizations, extensions and restrictions. Serveral concrete examples of practical applications in functional analysis, geometry, probability and statistics, physics and mathematical imaging are detailed.
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Read the articleAUTHOR
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Jean-Charles PINOLI: Professor - École Nationale Supérieure des Mines de Saint-Etienne
INTRODUCTION
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.
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KEYWORDS
Lebesgue spaces | Sobolev spaces | Caccioppoli sets | trace operator | Radon measures
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