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Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit. Chain rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behaviors are described by functions or functionals with a very limited degree of smoothness.

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The following chain rule is proved in the paper Vol'pert , p. Note all partial derivatives must be interpreted in a generalized sense, i. It is possible to generalize the above notion of total variation so that different variations are weighted differently. SBV functions i. Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper De Giorgi contains a useful bibliography. The space of all sequences of finite total variation is denoted by bv.

The norm on bv is given by. Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. For real functions of several real variables. The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation.

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Also there is some modern application which deserves a brief description. From Wikipedia, the free encyclopedia. Brudnyi, Yuri , "Multivariate functions of bounded k , p —variation" , in Randrianantoanina, Beata; Randrianantoanina, Narcisse eds. Includes a discussion of the functional-analytic properties of spaces of functions of bounded variation. A good reference on the theory of Caccioppoli sets and their application to the minimal surface problem. The link is to a preview of a later reprint by Springer-Verlag.

The whole book is devoted to the theory of BV functions and their applications to problems in mathematical physics involving discontinuous functions and geometric objects with non-smooth boundaries. Maybe the most complete book reference for the theory of BV functions in one variable: classical results and advanced results are collected in chapter 6 " Bounded variation " along with several exercises. The first author was a collaborator of Lamberto Cesari.

Intermediate and extrapolated spaces for bi-continuous operator semigroups

Kolmogorov, Andrej N. One of the most complete monographs on the theory of Young measures , strongly oriented to applications in continuum mechanics of fluids. Maz'ya, Vladimir G. One of the best monographs on the theory of Sobolev spaces. In this paper, Musielak and Orlicz developed the concept of weighted BV functions introduced by Laurence Chisholm Young to its full generality.

A seminal paper where Caccioppoli sets and BV functions are thoroughly studied and the concept of functional superposition is introduced and applied to the theory of partial differential equations : it was also translated in English as Vol'Pert, A I , "Spaces BV and quasi-linear equations", Mathematics USSR-Sbornik , 2 2 : —, doi : The normalised positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions.

This point of view has been important in spectral theory , [4] in particular in its application to ordinary differential equations.


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Functions of bounded variation, BV functions , are functions whose distributional derivative is a finite [5] Radon measure. More precisely:. Definition 2. Sometimes, especially in the theory of Caccioppoli sets , the following notation is used. Hence the continuous linear functional defines a Radon measure by the Riesz-Markov Theorem. If the function space of locally integrable functions , i. Precisely, developing this idea for definition 2.

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The second one, which is adopted in references Vol'pert and Maz'ya partially , is the following:. Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and proofs will be carried on only for functions of several variables since the 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 Giusti , pp. The quantities. Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit. Chain rules for nonsmooth functions are very important in mathematics and mathematical physics since there are several important physical models whose behaviors are described by functions or functionals with a very limited degree of smoothness.

The following chain rule is proved in the paper Vol'pert , p. Note all partial derivatives must be interpreted in a generalized sense, i.


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It is possible to generalize the above notion of total variation so that different variations are weighted differently. SBV functions i. Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper De Giorgi contains a useful bibliography.

The space of all sequences of finite total variation is denoted by bv. The norm on bv is given by.

Part I: Complex Variables, Lec 2: Functions of a Complex Variable

Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. For real functions of several real variables. The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation.

Also there is some modern application which deserves a brief description.

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