Vector‐valued optimal Lipschitz extensions

Consider a bounded open set $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}U \subset \R^n$ and a Lipschitz function $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}g: \partial U \to \R^m$. Does this function always have a canonical optimal Lipschitz extension to all of U? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case n = m = 2, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs. © 2011 Wiley Periodicals, Inc.

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