A-Quasiconvexity, Lower Semicontinuity, and Young Measures

The notion of A-quasiconvexity is introduced as a necessary and sufficient condition for (sequential) lower semicontinuity of (u,v) ↦ ∫Ω f(x,u(x), v(x))\, dx whenever f: Ω× Rᵐ× Rᵈ → [0,+∞) is a normal integrand, Ω⊂ RN is open, bounded, uₙ → u in measure, vₙ v in Lᵖ(Ω; Rᵈ) ( if p = +∞), and A vₙ → 0 in W-1,p(Ω) (A vₙ = 0 if p=+∞). Here Av = ∑ᵢ₌₁N A⁽ⁱ⁾ ∂ v ∂ xᵢ is aconstant rank partial differential operator, A⁽ⁱ⁾ ∈ ( Rᵈ; Rˡ), and f(x,u,·) is A-quasi-convex if f(x,u,v) ≤ ∫Q f(x,u,v + w(y)) \, dy for all v ∈ Rᵈ and all w ∈ C∞(Q; Rᵈ) such that A w=0, ∫Q w(x) \, dx = 0, and w is Q-periodic, Q := (0,1)N. The characterization of Young measures generated by such sequences ₙ\ is obtained for 1 ≤ p < + ∞, thus recovering the well-known results for the framework A= curl, i.e., when vₙ = ∇ φₙ for some φₙ ∈ W1,p(Ω; Rᵐ), d = N × m. In this case A-quasiconvexity reduces to Morrey's notion of quasiconvexity.