Slope estimate and boundary differentiability of infinity harmonic functions on convex domains

We study the boundary differentiability of infinity harmonic functions with given differentiable boundary data on convex domains. At a flat point (the boundary point where the blow-up of the domain is the half-space), the infinity harmonic function uu is differentiable due to a previous result of the first author in Hong (2013). At a corner point (the boundary point where the blow-up of the domain is not the half-space), an example shows that uu is not necessarily differentiable. In this paper, we establish a slope estimate for uu at corner points and provide a necessary and sufficient condition for the differentiability of uu at corner points.