Quasiconvexity in the Riemannian setting

TL;DR

This paper generalizes the concept of quasiconvexity to Riemannian manifolds and proves its equivalence to the lower semicontinuity of certain integral functionals in this setting.

Contribution

It introduces a Riemannian quasiconvexity notion and establishes its role in characterizing lower semicontinuity of integral functionals on manifolds.

Findings

Riemannian quasiconvexity generalizes Euclidean quasiconvexity.

The introduced condition characterizes lower semicontinuity of integral functionals.

The results hold for functions defined on vector bundles over Riemannian manifolds.

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 \[ F(u, \Omega) = \int_{\Omega} f(du) \, d\mu \] with respect to the weak$^*$ topology of $W^{1,\infty}(\Omega, \mathbb{R}^m)$, for every bounded open subset $\Omega\subseteq M$.