Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A generalized Stoilow decomposition for pairs of mappings of integrable
  dilatation   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 $Du(z)^T Du(z)=\lambda Dv(z)^T Dv(z)$ 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
$Du(z)=\mathcal{P}(\psi(w(z)))Dv(z)$ where
$\mathcal{P}(a+ib)=(\begin{smallmatrix} a & -b \\ b & a \end{smallmatrix})$.
  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.

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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex