*Accepted Paper*

**Inserted:** 13 nov 2012

**Last Updated:** 18 feb 2014

**Journal:** Advances in Calculus of Variations

**Year:** 2012

**Abstract:**

We prove a rigidity result for pairs of mappings of integrable dilatation whose gradients pointwise deform the unit ball to similar ellipses. Our result implies as corollaries a version of the generalized Stoilow decomposition provided by Theorem 5.5.1 of a recent monograph of Astala-Iwaniec-Martin and the two dimensional rigidity result of our previous paper for mappings whose symmetric part of gradient agrees.

Specifically let $u,v\in W^{1,2}(\Omega,\mathbb{R}^2)$ where $\det(Du)>0$, $\det(Dv)>0$ a.e. and $u$ is a mapping of integrable dilatation. Suppose for a.e. $z\in \Omega$ we have $S(Du(z))=\lambda S(Dv(z))$ for some $\lambda>0$. Then there exists a meromorphic function $\psi$ and a homeomorphism $w\in W^{1,1}(\Omega:\mathbb{R}^2)$ such that $Du(z)=\mathcal{P}\left(\psi(w(z))\right)Dv(z)$ where $\mathcal{P}(a+ib)=\left(\begin{smallmatrix} a & -b \\ b & a\end{smallmatrix}\right)$.

We show by example that this result is sharp in the sense that there can be no continuous relation between the gradients of $Du$ and $Dv$ on a dense open connected subset of $\Omega$ unless one of the mappings is of integrable dilatation.

**Download:**