A $C^1$-regularity result to the inhomogeneous normalized infinity Laplacian equation

Abstract

In this paper, we investigate the regularity for the viscosity solution to the Dirichlet problem
\begin{equation*}
\left\{
\begin{array}{ll}
-\Delta_\infty^N u = f(x)\quad &{\rm in}\;\Omega,\\
u=0 \quad &{\rm on}\;\partial\Omega,
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded convex domain and $f(x)\in C (\Omega)$.
For $0<f_{\inf}=\inf_\Omega f\leq f \leq \sup_\Omega f=f_{\sup}<+\infty$, we first prove the $\frac{1}{2}$-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 $-\infty<f_{\inf}\leq f \leq f_{\sup}<0$.