Vector-valued optimal Lipschitz extensions

Details Vector-valued optimal Lipschitz extensions Vector-valued optimal Lipschitz extensions

2010-06-09

arxiv.org/abs/1006.1741v1

Mathematics

Analysis of PDEs

24 pages, 10 figures

Scientific paper

Consider a bounded open set $U$ in $R^n$ and a Lipschitz function g from the boundary of $U$ 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.

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