C1,α-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$u_t=\left|Du{|}^{\gamma }F\right(D^{2}u)+f\text{in }{Q}_1,$ where $\gamma >-1$ and $f\in C\left(Q_1\right)\cap L^{\infty }\left(Q_1\right)$. For this purpose, we prove the well-posedness of the regularized Cauchy-Dirichlet problem$\{\begin{array}{cccc}{u}_t& ={(1+|Du{|}^2)}^{\gamma /2}F\left(D^{2}u\right)& & \text{in }{Q}_1\\ u& =\phi & & \text{on }{\partial }_p{Q}_1,\end{array}$ where $\gamma >-2$. Our approach utilizes the Bernstein method with approximations in view of the difference quotient.