A Note on Meyers’ Theorem in W

Lower semicontinuity properties of multiple integrals u ∈W (Ω;R) 7→ ∫ Ω f(x, u(x), · · · ,∇u(x)) dx are studied when f may grow linearly with respect to the highest-order derivative, ∇ku, and admissible W k,1(Ω;Rd) sequences converge strongly in W k−1,1(Ω;Rd). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ 7→ f(x0, u(x0), · · · ,∇u(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)).