A-quasi-convexity with variable coefficients

Abstract

It is shown that, for integrals of the type with Ω R^N open, bounded and f: Ω × R^m × R^d → [0, + ∞) Carathéodory satisfying a growth condition 0 ≤ f(x, u, υ) ≤ C(1 + |υ|^p), for some p ∈ (1, + ∞), a sufficient condition for lower semi-continuity along sequences u_n → u in measure, υ_n → υ in L^p, Aυ_n → 0 in W^{−1, p} is the A_x-quasi-convexity of f(x, u, ·). Here, A is a variable coefficients operator of the form with A(ⁱ) ∈ C^∞ (Ω; M^{l × d}) ∩ W^{1, ∞}, i = 1, …, N, satisfying the condition and A_x denotes the constant coefficients operator one obtains by freezing x. Under additional regularity conditions on f, it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences {υ_n} in L^p satisfying the condition Aυ_n → 0 in W^−1,p, is obtained.