C1-boundary regularity of planar infinity harmonic functions

Abstract

We prove that if ω C R2 is a bounded domain with C 2-boundary and g ε C2(R2), then any viscosity solution u ε C(ω) of the infinity Laplacian equation (1.1) is C1(ω). The interior C1 and C1,α-regularity of u in dimension two has been proved by Savin [20], and Evans and Savin [15], respectively. We also show that for any n ≥ 3, if ω ε R n is a bounded domain with C1-boundary and g ε C 1(Rn), then the solution u of equation (1.1) is differentiable on ?ω. This can be viewed as a supplementary result to the much deeper interior differentiability theorem by Evans and Smart [16, 17]. © 2012 International Press.