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N→∞ k→∞ Thus assume for contradiction that that there exists εn → 0+ and v ∈ Lq (D; Rd ) ∩ ker A such that Fhom (v; D) > Γ − lim inf Fεn (v; D). n→∞ q d v in Lq (D; Rd ) and Let {vn } ⊂ L (D; R ) ∩ ker A be such that vn Fhom (v; D) > lim inf Fεn (vn ; D), n→∞ and choose a subsequence {εnk } such that Fhom (v; D) > lim Fεnk (vnk ; D). k→∞ Then, by the previous lemmas, we can extract a further subsequence {εnkj } such that Fhom (v; D) = Γ − lim inf Fεnk (v; D) ≤ lim Fεnk (vnkj ; D) < Fhom (v; D), j→∞ j→∞ j j which is a contradiction and proves the claim.

Q Consequently dF − (v; ·) (x0 ) ≥ lim inf k→∞ dLN f (sk y, v(x0 ) + wk (y)) dy Q ≥ lim inf lim inf i→∞ k→∞ f (sk y, v(x0 ) + θi (y)wk (y)) dy, Q 30 where 0 ≤ θi ≤ 1 are smooth cut-off functions with compact support in Q such that θi ≡ 1 in (1 − 1/i)Q, and where we used the q-equi-integrability of {wk } and (A2 ). Then θi wk 0 in Lq (Q; Rd ) as k → ∞ and i → ∞, in this order, and lim lim A(θi wk ) = 0 in W −1,q (Q; Rl ). i→∞ k→∞ Diagonalize to get Ui := θi wki extended by zero outside Q, such that Ui AUi → 0 in W −1,q (Q; Rl ) as i → ∞, and lim inf lim inf i→∞ Thus k→∞ f (sk y, v(x0 ) + θi (y)wk (y)) dy = lim inf i→∞ Q 1 dF − (v; ·) (x0 ) ≥ lim inf N i→∞ s dLN ki ≥ lim inf i→∞ x, v(x0 ) + Ui s ki Q 1 sN ki − lim sup i→∞ f f f (ski , v(x0 ) + Ui (y)) dy.

Naz. Sci. XL 104 (1986), pp. 261–281. [9] A. Braides Relaxation of functionals with constraints on the divergence, Ann. Univ. , Sez. VII 33 (1987), pp. 157–177. [10] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Clarendon Press, Oxford, 1998. [11] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free–discontinuity problems, Arch. Rat. Mech. , 135 (1996), pp. 297–356. [12] G. Buttazzo, Semicontinuity, relaxation and integral representation problems in the Calculus of Variations, Pitman Res.

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