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.
Comparison principle to the infinity Laplacian equation with lower term
