2010
DOI: 10.1512/iumj.2010.59.4249
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Multiple integrals under differential constraints: two-scale convergence and homogenization

Abstract: Two-scale techniques are developed for sequences of mapssatisfying a linear dierential constraint Au k = 0. These, together with Γ-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type

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Cited by 28 publications
(48 citation statements)
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“…Homogenization in the context of variational problems restricted to A-free fields was first studied by Braides, Fonseca & Leoni in [9]. Later, this result was enhanced in [15], where the use of two-scale techniques allowed for weaker assumptions on the integrand.…”
Section: Collection Of Results On Homogenizationmentioning
confidence: 99%
“…Homogenization in the context of variational problems restricted to A-free fields was first studied by Braides, Fonseca & Leoni in [9]. Later, this result was enhanced in [15], where the use of two-scale techniques allowed for weaker assumptions on the integrand.…”
Section: Collection Of Results On Homogenizationmentioning
confidence: 99%
“…Due to the non-convexity of the sets M s and SO (2), it does not fall within the scope of works on gradient-constraint problems like [7,8,12], either. For a study of homogenization problems that involve constraints imposed by special linear first order partial differential equations we refer to [5,18,21].…”
Section: Introductionmentioning
confidence: 99%
“…Let us emphasize that variational problems with differential constraints naturally appear in hyperelasticity, electromagnetism, or in micromagnetics [7,25,26] and are closely related to the theory of compensated compactness [24,28,29]. The concept of A-quasiconvexity goes back to [5] and has been proved to be useful as a unified approach to variational problems with differential constraints, including results on homogenization [4,11], dimension reduction [19] and characterization of generalized Young measures [2] in the A-free setting. Moreover, first results on A-quasiaffine functions and weak continuity appeared in [16].…”
Section: Introductionmentioning
confidence: 99%