A note on Meyers' theorem in Wk,1

Lower semicontinuity properties of multiple integrals u ∈ Wk,1(Ω;ℝd) → ∫Ω f (x,u(x), ⋯, ∇ku(x)) dx are studied when f may grow linearly with respect to the highest-order derivative, ∇ku, and admissible Wk,1(Ω;ℝd) sequences converge strongly in Wk-l,1(Ω;ℝd). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ → f(x0, u(x0), ⋯, ∇k-1u(x0), ξ) ensures lower semicontinuity. The case where f(x0, u(x0), ⋯ , ∇k-1u(x0),·) is k-quasiconvex remains open except in some very particular cases, such as when f(x, u(x), ⋯ , ∇ku(x)) = h(x)g(∇ku(x)).