$C^{1, \alpha}$-regularity for solutions of degenerate/singular fully nonlinear parabolic equations

Abstract

We establish the interior $C^{1,\alpha}$-estimate for viscosity solutions of degenerate/singular fully nonlinear parabolic equations $$u_t = |Du|^{\gamma}F(D^2u) + f.$$ For this purpose, we prove the well-posedness of the regularized Dirichlet problem \begin{equation*} \left\{ \begin{aligned} u_t&=(1+|Du|^2)^{\gamma/2}F(D^2u) &&\text{in $Q_1$} \newline u&=\varphi &&\text{on $\partial_p Q_1$}. \end{aligned}\right. \end{equation*} Our approach utilizes the Bernstein method with approximations in view of difference quotient.