Partial Regularity for constrained minimisers of quasi convex functionals with $p$-growth

We consider minimisers of integral functionals $F$ over a ‘constrained’ class of $W^{1,p}$-mappings from a bounded domain into a compact Riemannian manifold $M$, i.e. minimisers of $F$ subject to holonomic constraints. Integrands of the form $f(Du)$ and the general $f(x,u,Du)$ are considered under natural strict $p$-quasiconvexity and $p$-growth assumptions for any exponent $1 < p <+\infty$. Unlike the harmonic and $p$-harmonic map case, the quasiconvexity condition requires one to linearise the map at the level of the gradient. In a bid to give a direct proof of partial $C^{1,α}-regularity for such minimisers, we developing an appropriate notion of a tangential harmonic approximation together with a discussion on the difficulties in establishing Caccioppoli-type inequalities. The need in the latter problem to construct suitable competitors to the minimiser via the so-called Luckhaus Lemma presents difficulties quite separate to that of the unconstrained case. We will give a proof of this lemma together with a discussion on the implications for higher integrability.