On multiwell Liouville theorems in higher dimension

Abstract

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 $p>1,q\ge 1$ and for connected ${\Omega }^{\prime }\subset \subset \Omega \subset {\mathbb{R}}^n$, there exists positive constant $a<1$ depending on $n,p,q,\Omega ,{\Omega }^{\prime }$ such that for $ \veps=\| {dist}(Du, K)\|_{L^p(Ω)}^p$ we have $\inf_{R\in K}\|Du-R\|^p_{L^p(Ω')}\leq M\veps^{1/p}$ provided $u$ satisfies the inequality $\| D^2 u\|_{L^q(Ω)}^q\leq a\veps^{1-q}$. 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 $n\ge 3$.

35 pages