*Published Paper*

**Inserted:** 6 feb 2008

**Last Updated:** 18 feb 2014

**Journal:** Advances in Calculus of Variations

**Volume:** 6

**Pages:** 247-298

**Year:** 2013

**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 $p>1, q \geq 1$ 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}\

Du-R\

^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

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