A remark on $C^{1,\alpha }$-regularity for differential inequalities in viscosity sense

Abstract

We prove interior $C^{1,\alpha }$-regularity for solutions \[ - Λ\leq F(D^2 u) \leq Λ\] where $\Lambda$ is a constant and $F$ is fully nonlinear, 1-homogeneous, uniformly elliptic. The proof is based on a reduction to the homogeneous equation $F(D^{2}u)=0$ by a blow-up argument -- i.e. just like what is done in the case of viscosity solutions $F(D^{2}u)=f$ for $f\in L^{\infty }$. However it was not clear to us that the above inequality implies $F(D^{2}u)=f$ for some bounded $f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on $C^{1,\alpha }$-regularity for viscosity inequalities. So we thought this result might be worth recording.

In the 'obvious' estimate in Theorem 1.2. there was an obvious mistake. Now its fixed, obviously