Tug-of-war and the infinity laplacian

Journal of the American Mathematical Society

Volumn 22, Issue 1, 2009, Pages 167-210

  Peres, Yuval ^a,b   Schramm, Oded ^a   Sheffield, Scott ^a,c   Wilson, David B ^a  

^a MICROSOFT RESEARCH   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c NEW YORK UNIVERSITY   (United States)

Author keywords

Absolutely minimal lipschitz extension; Infinity laplacian; Tug of war

Indexed keywords

EID: 65249085295     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-08-00606-1     Document Type: Article

References (25)

1. Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551-561. MR0217665 (36:754)
2. yy = 0, Ark. Mat. 7 (1968), 395-425. MR0237962 (38:239)
3. -, Construction of singular solutions to the p-harmonic equation and its limit equation for p = ∞, Manuscripta Math. 56 (1986), no. 2, 135-158. MR850366 (87j:35070)
4. Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N. S.) 41 (2004), no. 4, 439-505 (electronic). MR2083637 (2005k:35159)
5. M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner, and P. E. Souganidis, Viscosity solutions and applications, Lecture Notes in Mathematics, vol. 1660, Springer-Verlag, Berlin, 1997, Lectures given at the 2nd C. I. M. E. Session held in Montecatini Terme, June 12-20, 1995, edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C. I. M. E. [C. I. M. E. Foundation]. MR1462698 (98b:35005)
6. G. Barles and Jérôme Busca, Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Comm. Partial Differential Equations 26 (2001), no. 11-12, 2323-2337. MR1876420 (2002k:35078)
7. p = f and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino 1989 (1991). Special issue on topics in nonlinear PDEs, 15-68. MR1155453 (93a:35049)
8. Thierry Champion and Luigi De Pascale, Principles of comparison with distance functions for absolute minimizers, J. Convex Anal. 14 (2007), no. 3, 515-541. MR2341302
9. M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13 (2001), no. 2, 123-139. MR1861094 (2002h:49048)
10. Michael G. Crandall and L. C. Evans, A remark on infinity harmonic functions, Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000) (San Marcos, TX), Electron. J. Differ. Equ. Conf., vol. 6, Southwest Texas State Univ., 2001, pp. 123-129 (electronic). MR1804769 (2001j:35076)
11. Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1-42. MR690039 (85g:35029)
12. Lawrence C. Evans and Yifeng Yu, Various properties of solutions of the infinity-Laplacian equation, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1401-1428. MR2180310 (2006g:35062)
13. Robert Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), no. 1, 51-74. MR1218686 (94g:35063)
14. Petri Juutinen, Absolutely minimizing Lipschitz extensions on a metric space, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 1, 57-67. MR1884349 (2002m:54020)
15. Andrew J. Lazarus, Daniel E. Loeb, James G. Propp, Walter R. Stromquist, and Daniel H. Ullman, Combinatorial games under auction play, Games Econom. Behav. 27 (1999), no. 2, 229-264. MR1685133 (2001f:91023)
16. Andrew J. Lazarus, Daniel E. Loeb, James G. Propp, and Daniel Ullman, Richman games, Games of No Chance (Richard J. Nowakowski, ed.), MSRI Publications, vol. 29, Cambridge Univ. Press, Cambridge, 1996, pp. 439-449. MR1427981 (97j:90101)
17. Donald A. Martin, The determinacy of Blackwell games, J. Symbolic Logic 63 (1998), no. 4, 1565-1581. MR1665779 (2000d:03118)
18. E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837-842. MR1562984
19. V. A. Mil'man, Absolutely minimal extensions of functions on metric spaces, Mat. Sb. 190 (1999), no. 6, 83-110. MR1719573 (2000j:41041)
20. Abraham Neyman and Sylvain Sorin (eds.), Stochastic games and applications, NATO Science Series C: Mathematical and Physical Sciences, vol. 570, Dordrecht, Kluwer Academic Publishers, 2003. MR2032421 (2004h:91004)
21. Adam M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74 (2005), no. 251, 1217-1230 (electronic). MR2137000 (2006h:65165)
22. Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson, Random-turn Hex and other selection games, Amer. Math. Monthly 114 (2007), no. 5, 373-387. MR2309980 (2008a:91039)
23. 1 regularity for infinity harmonic functions in two dimensions, Arch. Ration. Mech. Anal. 176 (2005), no. 3, 351-361. MR2185662 (2006i:35108)
24. Hassler Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63-89. MR1501735
25. Yifeng Yu, Uniqueness of values of Aronsson operators and applications to "tug-of-war" game theory, 2007, http://www.ma.utexas.edu/~yifengyu/ uofa2d.pdf.