The Boundary Regularity for Normalized Infinity Laplace Equations

Abstract

Infinity Laplace equations, which derive from minimal Lipschitz extensions and absolutely minimal variational problems, have been widely applied in zero-sum tug-of-war game, optimal transport, shape deformation and so on. However, due to the quasi-linearity, extreme degeneration (non-degeneration only in the gradient direction) and non-divergence of the equations, it is difficult to define its classical or weak solutions. After introducing the idea of viscosity solutions, the theoretical research of infinite Laplace equations begin to develop. We study the boundary Hölder regularity of solutions for inhomogeneous normalized infinite Laplace equations on bounded domains. Main ideas are as follows: Firstly, we get bounded estimate of solutions through constructing barrier functions about super(sub)-solutions. Secondly, we use iterative method to approach the solutions of equations. Finally, we obtain regularity estimates near the boundary by calculating error between barrier functions and solutions of equations. This paper proves that the visvosity solutions of inhomogeneous normalized infinite Laplace equations are Hölder continious on Lipschitz boundary provided that the region boundary is Lipschitz continuous, the right inhomogeneous term is positive (negative) continuous and the boundary values are Hölder continuous. On the basis of our conclusion, the global Hölder regularity theory of the normalized infinite Laplace equations can be obtained combining with the internal regularity estimates. In addition, this method can be extended to the boundary estimates of the infinite fractional Laplace equations.