# On multiwell Liouville theorems in higher dimension - Mathematics > Classical Analysis and ODEs

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Abstract: We consider certain subsets of the space of $n\times n$ matrices of the form$K = \cup {i=1}^m SOnA i$, and we prove that for $p>1, q \geq 1$ and forconnected $\Omega-\subset\subset\Omega\subset \R^n$, there exists positiveconstant $a<1$ depending on $n,p,q, \Omega, \Omega-$ such that for $\veps=\|{dist}Du, K\| {L^p\Omega}^p$ we have $\inf {R\inK}\|Du-R\|^p {L^p\Omega-}\leq M\veps^{1-p}$ provided $u$ satisfies theinequality $\| D^2 u\| {L^q\Omega}^q\leq a\veps^{1-q}$. Our main result holdswhenever $m=2$, and also for {\em generic} $m\le n$ in every dimension $n\ge3$, as long as the wells $SOnA 1, ., SOnA m$ satisfy a certainconnectivity condition. These conclusions are mostly known when $n=2$, and theyare new for $n\ge 3$.

Autor: Robert L. Jerrard, Andrew Lorent

Fuente: https://arxiv.org/