Quasiconvexity and Lower Semicontinuity in the Riemannian Setting

We introduce a notion of quasiconvexity for integrands defined on the bundle of linear maps between the tangent spaces of two smooth, connected and complete Riemannian manifolds $(M,g)$ and $(N,h)$, naturally extending the classical Euclidean notion. For continuous integrands, we prove that this Riemannian quasiconvexity is equivalent to the sequential lower semicontinuity of the associated functional F(u,Ω)=∫_Ωf(du)\,dμg with respect to the weak^* convergence in W1,∞(Ω,N), in the intrinsic sense adopted in the paper, for every bounded open subset Ω⊆ M. We then establish the Riemannian analogues of the Acerbi--Fusco theorems for Carathéodory integrands satisfying the corresponding growth conditions, for every 1 p+∞, with respect to the weak and weak^* notions of convergence considered here.