Pointwise boundary differentiability for the infinity Laplace equations

In this paper, we study the boundary differentiability for infinity harmonic functions at some general corner points. We prove the slope estimate and provide a necessary and sufficient condition of the boundary differentiability for infinity harmonic function at less thanπcorner points or greater thanπcorner points (as defined in the Introduction). The boundary differentiable conclusions of the homogeneous infinity Laplace equations also apply to the inhomogeneous case.