A planar bi-Lipschitz extension theorem
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Sara Daneri
and
Aldo Pratelli
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.
Funding source: ERC
Award Identifier / Grant number: Starting Grant No. 258685
Funding source: ERC
Award Identifier / Grant number: Advanced Grant No. 226234
We warmly thank Tapio Rajala for some fruitful discussions and for having pointed out to us the paper [Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49–72]. We also want to thank the anonymous referees for their very careful work on this paper and their many precious suggestions. Part of this work was conceived while the first author was postdoc at the University of Pavia and at the University of Zurich. She would like to thank both Institutes for all the support received.
© 2015 by De Gruyter
