A C¹-regularity result to the inhomogeneous normalized infinity Laplacian equation

In this paper, we investigate the regularity for the viscosity solution to the Dirichlet problem {-ΔN∞u = f (x) in Ω, u = 0 on ∂Ω,, where Ω is a bounded convex domain and ƒ (x) ∈ C(Ω). For 0 ;lt {finf} = inf_Ω ƒ ≤ ƒ ≤ sup_Ω ƒ = {fₛᵤₚ} ;lt +∞, we ƒirst prove the 1/2-concavity of the viscosity solution by the convex envelope method of Alvarez-Lasry-Lions, and then establish the C¹ -regularity based on the upper estimate of semiconcave functions at the singular point. The similar result holds for-∞ :lt {finf} ≤ ƒ ≤ f_sup < 0.