A note on Meyers’ Theorem in $W^{k,1}$

Abstract

Lower semicontinuity properties of multiple integrals u ∈ W k,1 (Ω; R d) → Ω f (x, u(x), · · · , k u(x)) dx are studied when f may grow linearly with respect to the highest-order derivative , k u, and admissible W k,1 (Ω; R d) sequences converge strongly in W k−1,1 (Ω; R d). It is shown that under certain continuity assumptions on f, convexity, 1-quasiconvexity or k-polyconvexity of ξ → f (x 0 , u(x 0), · · · , k−1 u(x 0), ξ) ensures lower semicontinuity. The case where f (x 0 , u(x 0), · · · , k−1 u(x 0), ·) is k-quasiconvex remains open except in some very particular cases, such as when f (x, u(x), · · · , k u(x)) = h(x)g(k u(x)).