On coercive variational integrals

It is well-known that sequential weak lower semicontinuity of a variational integral (u, Ω) = ∫Ω F (∇u(x)) dx on the Sobolev space W^1,p (Ω, ℝ^N) under a p-growth condition on the integrand F is equivalent to quasiconvexity in the sense of Morrey. We show that coercivity on Dirichlet classes likewise is equivalent to a quasiconvexity condition. We also discuss more general notions of coercivity, and in the case of positively p-homogeneous integrands F we establish the existence of minimizers for a class of non-coercive quasiconvex variational integrals.