$C^{1+\alpha }$-regularity of viscosity solutions of general nonlinear parabolic equations

We investigate the $C^{1+\alpha }$-regularity of solutions of parabolic equations ${\mathrm{\partial }}_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_p$-viscosity solutions which are in $C_{loc}^{1+\alpha }$. We allow $H$ to be just measurable in $t$ and for its principal part to have sufficiently small discontinuities as a function of~$x$. No Lipschitz continuity of $H$ with respect to $v,Dv$ is required.