Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sobolev homeomorphisms and Brennan's conjecture   Let $\Omega \subset \mathbb{R}^n$ be a domain that supports the
$p$-Poincar\'e inequality. Given a homeomorphism $\varphi \in L^1_p(\Omega)$,
for $p>n$ we show the domain $\varphi(\Omega)$ has finite geodesic diameter.
This result has a direct application to Brennan's conjecture and quasiconformal
homeomorphisms. {\bf The Inverse Brennan's conjecture} states that for any
simply connected plane domain $\Omega' \subset\mathbb C$ with nonempty boundary
and for any conformal homeomorphism $\varphi$ from the unit disc $\mathbb{D}$
onto $\Omega'$ the complex derivative $\varphi'$ is integrable in the degree
$s$, $-2<s<2/3$. If $\Omega'$ is bounded than $-2<s\leq 2$. We prove that
integrability in the degree $s> 2$ is not possible for domains $\Omega'$ with
infinite geodesic diameter.

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The categories of this paper are: hep-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol