A P ] 2 8 A pr 2 00 6 Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u : X → R for which LipUu = Lip∂Uu for all open U ⊂ X r Y . This was previously known only for bounded domains in Rn, in which case u is infinity-harmonic, that is, a viscosity solution to ∆∞u = 0, where

[1]  Yuval Peres,et al.  Random-Turn Hex and Other Selection Games , 2005, Am. Math. Mon..

[2]  Editors , 1986, Brain Research Bulletin.

[3]  Adam M. Oberman.  A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions , 2004, Math. Comput..

[4]  G. Aronsson.  Extension of functions satisfying lipschitz conditions , 1967 .

[5]  G. Aronsson.  Construction of singular solutions to the p-harmonic equation and its limit equation for p=∞ , 1986 .

[6]  Donald A. Martin,et al.  The determinacy of Blackwell games , 1998, Journal of Symbolic Logic.

[7]  L. Evans,et al.  Optimal Lipschitz extensions and the infinity laplacian , 2001 .

[8]  V. A. Mil'man.  Absolutely minimal extensions of functions on metric spaces , 1999 .

[9]  R. Jensen.  Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient , 1993 .

[10]  Champion,et al.  Principles of comparison with distance functions for absolute minimizers , 2007 .

[11]  G. Barles,et al.  EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM* , 2001 .

[12]  Petri Juutinen,et al.  ABSOLUTELY MINIMIZING LIPSCHITZ EXTENSIONS ON A METRIC SPACE , 2002 .

[13]  J. Propp,et al.  Combinatorial Games under Auction Play , 1999 .

[14]  M. Crandall,et al.  A TOUR OF THE THEORY OF ABSOLUTELY MINIMIZING FUNCTIONS , 2004 .

[15]  Sylvain Sorin,et al.  Stochastic Games and Applications , 2003 .

[16]  L. Evans,et al.  Various Properties of Solutions of the Infinity-Laplacian Equation , 2005 .

[17]  Yifeng Yu.  A remark on infinity-harmonic functions. , 2006 .

[18]  Y. Peres,et al.  Tug-of-war and the infinity Laplacian , 2006, math/0605002.

[19]  J. Propp,et al.  Richman games , 1995, math/9502222.

[20]  E. J. McShane,et al.  Extension of range of functions , 1934 .

[21]  H. Whitney.  Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .