Sharp regularity estimates for second order fully nonlinear parabolic equations

Abstract

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form \begin{equation}\label{Meq}\tag{Eq} u_t- F(D^2u, Du, X, t) = f(X,t) \quad \mbox{in} \quad Q_1, \end{equation} where $F$ is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $\kappa(n, p, q):=\frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of \eqref{Meq} belongs. We prove that when $1< \kappa(n,p,q) < 2-\epsilon_F$, solutions are parabolic-H\"{o}lder continuous for a sharp, quantitative exponent $0< \alpha(n,p,q) < 1$. Precisely at the critical borderline case, $\kappa(n,p,q)= 1$, we obtain sharp Log-Lipschitz regularity estimates. When $0< \kappa(n,p,q) <1$, solutions are locally of class $C^{1+ \sigma, \frac{1+ \sigma}{2}}$ and in the limiting case $\kappa(n,p,q) = 0$, we show $C^{1, \text{Log-Lip}}$ regularity estimates provided $F$ has "better" \textit{a priori} estimates.