Sharp regularity estimates for second order fully nonlinear parabolic equations

TL;DR

This paper establishes sharp regularity estimates for solutions of fully nonlinear parabolic equations, identifying precise conditions under which solutions are Hölder, Log-Lipschitz, or differentiable with Log-Lipschitz derivatives.

Contribution

It provides the first sharp regularity results for viscosity solutions of fully nonlinear parabolic equations based on integrability conditions of the source term.

Findings

Solutions are Hölder continuous with a sharp exponent when 1<κ(n,p,q)<2-ε_F.

At the critical case κ(n,p,q)=1, solutions are Log-Lipschitz continuous.

For 0<κ(n,p,q)<1, solutions are locally C^{1+σ, (1+σ)/2} regular.

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.