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Mech. , 86 (1984), pp. 125 –145. [2] M. Amar and V. De Cicco, Relaxation of quasi-convex integrals of arbitrary order, Proc. Royal Soc. Edinburgh, 124A (1994), pp. 927–946. [3] L. Ambrosio, S. M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, J. Math. , 70 (1991), pp. 269–323. [4] E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. , 22 (1984), pp. 570–598. [5] J. M. Ball, A version of the fundamental theorem for Young measures, in PDE’s and Continuum Models of Phase Transitions, M.

Q x ski f dx x ski x, v(x0 ) + Ui ([ski ]+1)Q 1 sN ki 0 in Lq (Q; Rd ), x, v(x0 ) + Ui ([ski ]+1)Q\ski Q dx x ski dx, where [ski ] denotes the integer part of ski . We claim that the last limit is zero. Indeed 1 sN ki f x ski x, v(x0 ) + Ui ([ski ]+1)Q\ski Q dx = ([sk ]+1) i Q\Q sk i f (ski y, v(x0 ) + Ui (y)) dy. Since ([ski ] + 1)/ski → 1, we have that ([ski ] + 1) Q\Q = ski ([ski ] + 1) ski N − 1 → 0, and thus the claim follows from the q-equi-integrability of {Ui } and (A2 ). Hence, setting ni := [ski ] + 1 ∈ N, mi := 1/ski , we obtain dF − (v; ·) 1 (x0 ) ≥ lim inf N i→∞ n dLN i = lim inf i→∞ f (x, v(x0 ) + Ui (mi x)) dx ni Q f (ni y, v(x0 ) + Ui (ni mi y)) dy..

Plenum, New York, 1991. [38] L. Tartar, Some remarks on separately convex functions, in Microstructure and Phase Transitions, D. Kinderlehrer, R. D. James, M. Luskin and J. L. , Vol. 54, IMA Vol. Math. , Springer-Verlag, 1993, pp. 191–204. [39] L. C. Young, Lectures on Calculus of Variations and Optimal Control Theory, W. B. Saunders, 1969.

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A-quasiconvexity relaxation and homogenization by Andres Braides

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