Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ?², Ω an open bounded domain in ?^m , Y = 0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ?^m × ?ⁿ → ? such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ?^m × ?ⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that