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 - <p style="text-align: justify;">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.</p>