Quasiconvexity and Lower Semicontinuity in the Riemannian Setting

Abstract

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 $$ F(u,\Omega)=\int_\Omega f(du)\,d\mu_g $$ with respect to the weak$^*$ convergence in $W^{1,\infty}(\Omega,N)$, in the intrinsic sense adopted in the paper, for every bounded open subset $\Omega\subseteq M$. We then establish the Riemannian analogues of the Acerbi--Fusco theorems for Carath\'eodory integrands satisfying the corresponding growth conditions, for every $1\leqslant p\leqslant+\infty$, with respect to the weak and weak$^*$ notions of convergence considered here.