Characterization of Two-Scale Gradient Young Measures and Application to Homogenization

Abstract

Abstract. This work is devoted to the study of two-scale gradient Young measures naturally arising in nonlinear elasticity homogenization problems. Precisely, a characterization of this class of measures is derived and an integral representation formula for homogenized energies, whose integrands satisfy very weak regularity assumptions, is obtained in terms of two-scale gradient Young measures.

A note on concentrations for integral two-scale problems

“…The proof of (i)-(iii) was obtained in Babadjian, Baía and Santos [4] (Theorem 1.1), while the proof of iv) relies on the ideas of Theorem 1.1 in Fonseca, Pedregal and Müller [20], where a characterization of varifolds (concentrations measures) associated to uniformly bounded sequences of gradients of W 1, p -functions, with p > 1, is obtained.…”

“…We start by recalling the notion of two-scale Young measure as well as a characterization of these kind of measures derived in Babadjian, Baía and Santos [4] that we use below.…”

“…The measure ν has been characterized by many authors when {a n } satisfies a certain constraint (see Bocea and Fonseca [10], Kinderlehrer and Pedregal [21], Fonseca and Müller [19], Santos [29] and Babadjian, Baía and Santos [4]). In many physical situations energy densities of the form { f (x, a n (x))} develop concentration effects so that equality (1) doesn't hold anymore.…”

“…In Babadjian, Baía and Santos [4] the authors characterized the measure ν ≡ {ν (x,y) }, what is called a two-scale gradient Young measure (for oscillations), in terms of a Jensen's inequality with test functions in the space…”

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

“…Homogenization of periodic structures via two-scale convergence was introduced in [2,43] to deal with many problems formulated in terms of partial differential equations and integral functionals in Lebesgue and Sobolev spaces and BV ones (see [3,5,12,13,23,24]. Indeed the importance of detecting the overall behaviour of a material which might include periodically distributed heterogeneties is very important in many applications, from nonlinear elasticity, mean-field games, to micro and ferromagnetic, conductivity, evolutions problems, polycrystals, discrete models, etc.…”