Vector-valued optimal Lipschitz extensions
Author(s)Sheffield, Scott Roger; Smart, Charles
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Consider a bounded open set U C R[superscript n] and a Lipschitz function g : aU → R[superscript 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.
Original manuscript June 9, 2010
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Communications on Pure and Applied Mathematics
Sheffield, Scott, and Charles K. Smart. “Vector-valued optimal Lipschitz extensions.” Communications on Pure and Applied Mathematics 65, no. 1 (January 28, 2012): 128-154.