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 f₈₍₅ = inf_Ω ƒ ≤ ƒ ≤ sup_Ω ƒ = fₒₔ ;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 C^{1 -regularity based on the upper estimate of semiconcave functions at the singular point. The similar result holds for-∞: lt f₈₍₅ ≤ ƒ ≤ fₛup < 0.
Dong et al. (Wed,) studied this question.