A weighted eigenvalue problem of the biased infinity Laplacian*This work was partially supported by National Natural Science Foundation of China (No. 11501292 and 11531005). (8th February 2021)

• Record Type: Journal Article
• Authors: Liu, Fang; Yang, Xiao-Ping
• Is Part Of: Nonlinearity. Volume 34:Number 2(2021)
• Page Start: 1197
• Page End: 1237
• Publication Date: 2021-02-08
• Subjects: biased infinity Laplacian -- viscosity solution -- principal eigenvalue -- comparison principle -- Harnack inequality -- Lipschitz regularity; Primary 35P30 -- 35J60 -- 35J70 -- 35K55; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/abd85d
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study a weighted eigenvalue problem of the β -biased infinity Laplacian operator arising from the β -biased tug-of-war. We characterize the principal eigenvalue by the comparison principle and show that β -biased infinity Laplacian operator possesses two principal eigenvalues, corresponding to a positive and a negative principal eigenfunction. When a parameter is less than the principal eigenvalue, certain existence and uniqueness results of the inhomogeneous equations related to this problem are established. As an application, we obtain the decay estimates for viscosity solutions of the parabolic problem associated to the β -biased infinity Laplacian. In the process, we also establish the Lipschitz regularity and Harnack inequality by barrier method.