Some properties of the ground states of the infinity Laplacian

In this paper, we derive several properties of infinity ground states which are ground states of the infinity Laplacian in the sense of Juutinen-Lindqvist-Manfredi [14]. We will give a sufficient condition of the domain such that the distance function is the unique infinity ground state up to some constant factor. Those sufficient domains include the annulus, the ball, the stadium, etc. Also, we show that if the domain is convex, then a variational infinity ground state is a viscosity solution of the infinity Laplacian equation in the subdomain where it is C-1. This generalizes a result in Juutinen-Lindqvist-Manfredi [15].