Quasiconvexity in the Riemannian setting

Abstract

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and ${\mathbb{R}}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, Ω) = \int_Ω f(du) \, dμ\] with respect to the weak${}^{\ast }$ topology of $W^{1,\infty }(\Omega ,{\mathbb{R}}^m)$, for every bounded open subset $\Omega \subseteq M$.