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

$$\begin{aligned} u_t - F\left( D^2u, Du, X, t\right) = f(X,t) \quad \text{ in } \quad Q_1, \end{aligned}$$

(Eq)

where F is elliptic with respect to the Hessian argument and $f \in L^{p,q}(Q_1)$. The quantity $\Xi (n, p, q) := \frac{n}{p}+\frac{2}{q}$ determines to which regularity regime a solution of (Eq) belongs. We prove that when $1< \Xi (n,p,q) < 2-\epsilon _F$, solutions are parabolically $\alpha$-Hölder continuous for a sharp, quantitative exponent $0< \alpha (n,p,q) < 1$. Precisely at the critical borderline case, $\Xi (n,p,q)= 1$, we obtain sharp parabolic Log-Lipschitz regularity estimates. When $0< \Xi (n,p,q) <1$, solutions are locally of class $C^{1+ \sigma , \frac{1+ \sigma }{2}}$ and in the limiting case $\Xi (n,p,q) = 0$, we show parabolic $C^{1, \text {Log-Lip}}$ regularity estimates provided F has “better” a priori estimates.