We prove that, given a planar bi-Lipschitz map u defined on the boundary of the unit square, it is possible to extend it to a function v of the whole square, in such a way that v is still bi-Lipschitz. In particular, denoting by L and L the bi-Lipschitz constants of u and v, with our construction one has L ≤ CL4 (C being an explicit geometric constant). The same result was proved in 1980 by Tukia (see [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49-72]), using a completely different argument, but without any estimate on the constant L. In particular, the function v can be taken either smooth or (countably) piecewise affine

A planar bi-Lipschitz extension theorem

Daneri S
2015

Abstract

We prove that, given a planar bi-Lipschitz map u defined on the boundary of the unit square, it is possible to extend it to a function v of the whole square, in such a way that v is still bi-Lipschitz. In particular, denoting by L and L the bi-Lipschitz constants of u and v, with our construction one has L ≤ CL4 (C being an explicit geometric constant). The same result was proved in 1980 by Tukia (see [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49-72]), using a completely different argument, but without any estimate on the constant L. In particular, the function v can be taken either smooth or (countably) piecewise affine
bi-Lipschitz, extension, piecewise affine
File in questo prodotto:
File Dimensione Formato  
2015_AdvCalcVar_8_Daneri.pdf

non disponibili

Descrizione: Articolo principale
Tipologia: Versione Editoriale (PDF)
Licenza: Non pubblico
Dimensione 901.06 kB
Formato Adobe PDF
901.06 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.12571/7408
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 14
social impact