A planar bi-Lipschitz extension Theorem

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.