6533b85efe1ef96bd12c09ea

RESEARCH PRODUCT

Sobolev homeomorphic extensions onto John domains

Pekka KoskelaAleksis KoskiJani Onninen

subject

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

description

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.

yearjournalcountryeditionlanguage
2020-04-20
https://dx.doi.org/10.48550/arxiv.2004.09669