TY - JOUR T1 - Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation AU - Liu , Fang AU - Meng , Fei AU - Chen , Xiaoyan JO - Analysis in Theory and Applications VL - 4 SP - 439 EP - 450 PY - 2023 DA - 2023/01 SN - 38 DO - http://doi.org/10.4208/ata.OA-2020-0002 UR - https://global-sci.org/intro/article_detail/ata/21358.html KW - $β$−biased infinity Laplacian, viscosity solution, exponential cone, Harnack inequality, Lipschitz regularity. AB -

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.