scholarly journals Existence of Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian

2015 ◽  
Vol 03 (05) ◽  
pp. 488-495
Author(s):  
Fang Liu
Keyword(s):  
Viscosity Solutions ◽  
Inhomogeneous Equation ◽  
Infinity Laplacian

Solutions to an inhomogeneous equation involving infinity Laplacian

2012 ◽  
Vol 75 (14) ◽  
pp. 5693-5701 ◽  
Author(s):  
Fang Liu ◽  
Xiao-Ping Yang
Keyword(s):  
Inhomogeneous Equation ◽  
Infinity Laplacian

Harnack inequality and an asymptotic mean-value property for the Finsler infinity-Laplacian

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Benyam Mebrate ◽  
Ahmed Mohammed
Keyword(s):  
Viscosity Solutions ◽  
Harnack Inequality ◽  
Mean Value ◽  
Infinity Laplacian ◽  
Laplace Operators ◽  
Mean Value Property ◽  
Viscosity Sense

Abstract In this paper, we prove a Harnack inequality for nonnegative viscosity supersolutions of nonhomogeneous equations associated with normalized Finsler infinity-Laplace operators. Viscosity solutions to homogeneous equations are also characterized via an asymptotic mean-value property, understood in a viscosity sense.

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

2019 ◽  
Vol 19 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Fang Liu ◽  
Feida Jiang
Keyword(s):  
Viscosity Solutions ◽  
Comparison Principle ◽  
Laplacian Operator ◽  
Inhomogeneous Term ◽  
Infinity Laplacian ◽  
Lipschitz Continuous ◽  
Laplacian Equation ◽  
The Game Theory ◽  
Fixed Constant ◽  
Dirichlet Boundary Problem

Abstract In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when {f=0} , we show some explicit solutions.

scholarly journals Viscosity solutions for a polymer crystal growth model

2011 ◽  
Vol 60 (3) ◽  
pp. 895-936
Author(s):  
Pierre Cardaliaguet ◽  
Olivier Ley ◽  
Aurelien Monteillet
Keyword(s):  
Crystal Growth ◽  
Growth Model ◽  
Viscosity Solutions ◽  
Polymer Crystal
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