We prove that if U\\subset\\R^n is an open domain whose closure \\overline{U} is compact in the path metric, and F is a Lipschitz function on \\partial{U}, then for each \\beta\\in\\R there exists a unique viscosity solution to the \\beta-biased infinity Laplacian equation \\beta |\\nabla u| + \\Delta_\\infty u=0 on U that extends F, where \\Delta_\\infty u= |\\nabla u|^{-2} \\sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}.
\nIn the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the \\beta-biased \\eps-game as follows. The starting position is x_0 \\in U. At the k^\\text{th} step the two players toss a suitably biased coin (in our key example, player I wins with odds of \\exp(\\beta\\eps) to 1), and the winner chooses x_k with d(x_k,x_{k-1}) < \\eps. The game ends when x_k \\in \\partial{U}, and player II pays the amount F(x_k) to player I. We prove that the value u^{\\eps}(x_0) of this game exists, and that \\|u^\\eps – u\\|_\\infty \\to 0 as \\eps \\to 0, where u is the unique extension of F to \\overline{U} that satisfies comparison with \\beta-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with \\beta-exponential cones if and only if it is a viscosity solution to the \\beta-biased infinity Laplacian equation.<\/p>\n","protected":false},"excerpt":{"rendered":"
We prove that if U\\subset\\R^n is an open domain whose closure \\overline{U} is compact in the path metric, and F is a Lipschitz function on \\partial{U}, then for each \\beta\\in\\R there exists a unique viscosity solution to the \\beta-biased infinity Laplacian equation \\beta |\\nabla u| + \\Delta_\\infty u=0 on U that extends F, where \\Delta_\\infty […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer-Verlag","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"Calculus of Variations and Partial Differential 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