Abstract
This chapter introduces the definition of boundary regular (game-regular) points and proves that boundary regularity implies uniform convergence of game values to the unique solution of the Dirichlet p-Laplace problem. Further, it establishes two sufficient conditions for game-regularity: exterior cone property and p surpassing the dimension N of the problem. The following topics are covered: relation of game-regularity and convergence, technique of concatenating strategies, the annulus walk estimate and construction of barriers when p > N.
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References
Y. Peres and S. Sheffield. Tug-of-war with noise: a game-theoretic view of the p-Laplacian. Duke Math J., 145: 91–120, 2008.
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Lewicka, M. (2020). Game-Regularity and Convergence: Case p ∈ (2, ∞). In: A Course on Tug-of-War Games with Random Noise. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-46209-3_5
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DOI: https://doi.org/10.1007/978-3-030-46209-3_5
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