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 [Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, 2009] and the two-dimensional rigidity result of [Calc. Var. Partial Differential Equations (2012), DOI 10.1007/s00526-012-0566-4] for mappings whose symmetric part of gradient agrees.

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

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 Ω unless one of the mappings is of integrable dilatation.