Sobolev Homeomorphisms and Brennan’s Conjecture

V. Gol’dshtein; A. Ukhlov

doi:10.1007/s40315-014-0065-z

Sobolev Homeomorphisms and Brennan’s Conjecture

V. Gol’dshtein, A. Ukhlov

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

Let (Formula presented.) be a domain that supports the (Formula presented.)-Poincaré inequality. Given a homeomorphism (Formula presented.), for (Formula presented.) we show that the domain (Formula presented.) has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain (Formula presented.) with non-empty boundary and for any conformal homeomorphism (Formula presented.) from the unit disc (Formula presented.) onto (Formula presented.) the complex derivative (Formula presented.) is integrable in the degree (Formula presented.). If (Formula presented.) is bounded then (Formula presented.). We prove that integrability in the degree (Formula presented.) is not possible for domains (Formula presented.) with infinite geodesic diameter.

Original language	English
Pages (from-to)	247-256
Number of pages	10
Journal	Computational Methods and Function Theory
Volume	14
Issue number	2-3
DOIs	https://doi.org/10.1007/s40315-014-0065-z
State	Published - 31 Oct 2014

Keywords

Brennan’s conjecture
Sobolev homeomorphisms

ASJC Scopus subject areas

Analysis
Computational Theory and Mathematics
Applied Mathematics

Access to Document

10.1007/s40315-014-0065-z

Cite this

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abstract = "Let (Formula presented.) be a domain that supports the (Formula presented.)-Poincar{\'e} inequality. Given a homeomorphism (Formula presented.), for (Formula presented.) we show that the domain (Formula presented.) has finite geodesic diameter. This result has a direct application to Brennan{\textquoteright}s conjecture and quasiconformal homeomorphisms. The Inverse Brennan{\textquoteright}s conjecture states that for any simply connected plane domain (Formula presented.) with non-empty boundary and for any conformal homeomorphism (Formula presented.) from the unit disc (Formula presented.) onto (Formula presented.) the complex derivative (Formula presented.) is integrable in the degree (Formula presented.). If (Formula presented.) is bounded then (Formula presented.). We prove that integrability in the degree (Formula presented.) is not possible for domains (Formula presented.) with infinite geodesic diameter.",

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T1 - Sobolev Homeomorphisms and Brennan’s Conjecture

AU - Gol’dshtein, V.

AU - Ukhlov, A.

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N2 - Let (Formula presented.) be a domain that supports the (Formula presented.)-Poincaré inequality. Given a homeomorphism (Formula presented.), for (Formula presented.) we show that the domain (Formula presented.) has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain (Formula presented.) with non-empty boundary and for any conformal homeomorphism (Formula presented.) from the unit disc (Formula presented.) onto (Formula presented.) the complex derivative (Formula presented.) is integrable in the degree (Formula presented.). If (Formula presented.) is bounded then (Formula presented.). We prove that integrability in the degree (Formula presented.) is not possible for domains (Formula presented.) with infinite geodesic diameter.

AB - Let (Formula presented.) be a domain that supports the (Formula presented.)-Poincaré inequality. Given a homeomorphism (Formula presented.), for (Formula presented.) we show that the domain (Formula presented.) has finite geodesic diameter. This result has a direct application to Brennan’s conjecture and quasiconformal homeomorphisms. The Inverse Brennan’s conjecture states that for any simply connected plane domain (Formula presented.) with non-empty boundary and for any conformal homeomorphism (Formula presented.) from the unit disc (Formula presented.) onto (Formula presented.) the complex derivative (Formula presented.) is integrable in the degree (Formula presented.). If (Formula presented.) is bounded then (Formula presented.). We prove that integrability in the degree (Formula presented.) is not possible for domains (Formula presented.) with infinite geodesic diameter.

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KW - Sobolev homeomorphisms

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DO - 10.1007/s40315-014-0065-z

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Sobolev Homeomorphisms and Brennan’s Conjecture

Abstract

Keywords

ASJC Scopus subject areas

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