On multiwell Liouville theorems in higher dimension

We consider certain subsets of the space of $n\times n$ matrices of the form $K = \cup_{i=1}^m SO(n)A_i$, and we prove that for $p>1, q \geq 1$ and for connected $\Omega'\subset\subset\Omega\subset \R^n$, there exists positive constant $a<1$ depending on $n,p,q, \Omega, \Omega'$ such that for $\veps=\| {dist}(Du, K)\|_{L^p(\Omega)}^p$ we have $\inf_{R\in K}\|Du-R\|^p_{L^p(\Omega')}\leq M\veps^{1/p}$ provided $u$ satisfies the inequality $\| D^2 u\|_{L^q(\Omega)}^q\leq a\veps^{1-q}$. Our main result holds whenever $m=2$, and also for {\em generic} $m\le n$ in every dimension $n\ge 3$, as long as the wells $SO(n)A_1,..., SO(n)A_m$ satisfy a certain connectivity condition. These conclusions are mostly known when $n=2$, and they are new for $n\ge 3$.