Search results for "Sobolev extensions"

showing 2 items of 2 documents

Bi-Sobolev extensions

2022

We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling-Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling-Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.

Sobolev extensionskvasikonformikuvauksetMathematics - Complex VariablesPrimary 46E35 30C62. Secondary 58E20FOS: Mathematicsharmonic extensionquasiconformal mapping and mapping of finite distortionSobolev homeomorphismsComplex Variables (math.CV)Beurling-Ahlfors extension
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Sobolev homeomorphic extensions onto John domains

2020

Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schoenflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous $W^{1,2}$-extension but not even a homeomorphic $W^{1,1}$-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents $p<2$. John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory.

funktioteoriaMathematics::Dynamical SystemsSobolev extensionsMathematics - Complex Variables46E35 58E20quasidisksFOS: MathematicsMathematics::General TopologySobolev homeomorphismsComplex Variables (math.CV)John domainsfunktionaalianalyysiMathematics::Geometric Topology
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