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 where , a.e. and u is a mapping of integrable dilatation. Suppose for a.e. we have Then there exists a meromorphic function ψ and a homeomorphism such that where
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.
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