Calculus of Variations and Geometric Measure Theory

R. L. Jerrard - A. Lorent

On multiwell Liouville theorems in higher dimension

created by lorent on 06 Feb 2008
modified on 18 Feb 2014


Published Paper

Inserted: 6 feb 2008
Last Updated: 18 feb 2014

Journal: Advances in Calculus of Variations
Volume: 6
Pages: 247-298
Year: 2013


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 $\Omega'\subset\subset\Omega\subset \mathbb{R}^n$, there exists positive constant $a<1$ depending on $n,p,q, \Omega, \Omega'$ such that for $\epsilon=\
\mbox{dist}(Du, K)\
_{L^p(\Omega)}^p$ we have $\inf_{R\in K}\
^p_{L^p(\Omega')}\leq M\epsilon^{1/p}$ provided $u$ satisfies the inequality $\
D^2 u\
_{L^q(\Omega)}^q\leq a\epsilon^{1-q}$. Our main result holds whenever $m=2$, and also for generic $m\le n$ in every dimension $n\ge 3$, as long as the wells $SO(n)A_1,\ldots, SO(n)A_m$ satisfy a certain connectivity condition. These conclusions are mostly known when $n=2$, and they are new for $n\ge 3$.

Keywords: rigidity, Muti Wells, Liouville