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A relaxation result in BV for integral functionals with discontinuous integrands
Boundary behaviour of -harmonic functions in domains beyond Lipschitz domains Weak lower semicontinuity for non coercive polyconvex integrals Self-improving properties of weighted Hardy inequalities On a nonlinear fourth order elliptic system with critical growth in first order derivatives. Weak lower semicontinuity for non coercive polyconvex integrals. Scarpa 16, Roma, Italy. E-mail: amar dmmm. E-mail: decicco dmmm.
Function of bounded variation
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About the article Received : Revised : Published Online : Published in Print : Citing Articles Here you can find all Crossref-listed publications in which this article is cited. DOI: Lower semicontinuity for polyconvex integrals without coercivity assumptions. Related Content Loading General note: By using the comment function on degruyter. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.
Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains Weak lower semicontinuity for non coercive polyconvex integrals Self-improving properties of weighted Hardy inequalities On a nonlinear fourth order elliptic system with critical growth in first order derivatives. Powered by PubFactory. Definition 1. If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,. Through the Stieltjes integral , any function of bounded variation on a closed interval [ a , b ] defines a bounded linear functional on C [ a , b ].
In this special case,  the Riesz representation theorem states that every bounded linear functional arises uniquely in this way. 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 ,  in particular in its application to ordinary differential equations. Functions of bounded variation, BV functions , are functions whose distributional derivative is a finite  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. 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. 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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- Function of bounded variation.
- 1. Introduction.
- A chain rule in and its applications to lower semicontinuity.
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.
themisanthropelondon.com/no-precio-zithromax-250mg.php 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.
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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. Kolmogorov, Andrej N.
One of the most complete monographs on the theory of Young measures , strongly oriented to applications in continuum mechanics of fluids.