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

We prove interior $C^{1, α}$ -regularity for solutions [

\Lambda \leq F(D^2 u) \leq \Lambda ] where $Λ$ 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 $f o r s o m e b o u n d e d$ f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on $C^{1, α}$ -regularity for viscosity inequalities. So we thought this result might be worth recording.

We prove interior $C^{1, α}$ -regularity for solutions [

\Lambda \leq F(D^2 u) \leq \Lambda ] where $Λ$ 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 $f o r s o m e b o u n d e d$ f$ (as would be the case for linear equations in distributional sense by approximation). Nor were we able to find the literature on $C^{1, α}$ -regularity for viscosity inequalities. So we thought this result might be worth recording.

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