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Extension of internally bilipschitz maps in John disks. (English) Zbl 1100.30003
A domain \(D \subset \mathbb R^2\) is called a \(c\)-John domain if any two points \(z_1, z_2 \in D\) may be joined by a rectifiable arc \(\alpha \subset D\) such that \[ \min_{j = 1, 2} \text{length}(\alpha[z_j, z]) \leq c \text{dist}(z, \partial D) . \] A \(c\)-John disk is a simply connected \(c\)-John domain with at least two boundary points. The internal distance between two points \(z_1, z_2 \in D\) is \(\lambda_D(z_1, z_2) = \inf \text{length}(\alpha)\), where the infimum is taken over all rectifiable arcs \(\alpha \subset D\) joining \(z_1\) and \(z_2\). A homeomorphism \(f \colon D \rightarrow D'\) is said to be internally \(L\)-bilipschitz if \[ \frac1L \lambda_D(z_1, z_2) \leq \lambda_{D'}(f(z_1), f(z_2)) \leq L \lambda_D(z_1, z_2) \] for all points \(z_1, z_2 \in D\). The symbol \(\partial_r D\) denotes the set of all rectifiably accesible points of \(\partial D\). After a discussion of internal distance on the boundary, a three point property for John disks is proved. The main theorem of the paper is a result about extension of internally bilipschitz maps in John disks, reminiscent of a result of Gehring for quasidisks; Suppose \(D\) is a Jordan \(c\)-John disk and that \(f \colon \partial D \rightarrow \partial D'\) is a homeomorphism. If \(f \colon (f^{-1}(\partial_r D'), \lambda_D) \rightarrow (\partial_r D', \lambda_{D'})\) is \(L\)-bilipschitz, then \(D'\) is a \(c'\)-John domain, and \(f\) extends as an \(M\)-bilipschitz map \(\tilde f \colon (\overline D, \lambda_D) \rightarrow (\overline{D'}, \lambda_{D'})\). \(M\) and \(c'\) depend only on \(L\) and \(c\).

30C20 Conformal mappings of special domains
30C62 Quasiconformal mappings in the complex plane
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