Sharp regularity estimates for second order fully nonlinear parabolic equations

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form Equt-FD2u,Du,X,t=f(X,t)inQ1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t - F\left( D^2u, Du, X, t\right) = f(X,t) \quad \text{ in } \quad Q_1, \end{aligned}$$\end{document}where F is elliptic with respect to the Hessian argument and f∈Lp,q(Q1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in L^{p,q}(Q_1)$$\end{document}. The quantity Ξ(n,p,q):=np+2q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi (n, p, q) := \frac{n}{p}+\frac{2}{q}$$\end{document} determines to which regularity regime a solution of (Eq) belongs. We prove that when 1<Ξ(n,p,q)<2-ϵF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< \Xi (n,p,q) < 2-\epsilon _F$$\end{document}, solutions are parabolically α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Hölder continuous for a sharp, quantitative exponent 0<α(n,p,q)<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< \alpha (n,p,q) < 1$$\end{document}. Precisely at the critical borderline case, Ξ(n,p,q)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi (n,p,q)= 1$$\end{document}, we obtain sharp parabolic Log-Lipschitz regularity estimates. When 0<Ξ(n,p,q)<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< \Xi (n,p,q) <1$$\end{document}, solutions are locally of class C1+σ,1+σ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1+ \sigma , \frac{1+ \sigma }{2}}$$\end{document} and in the limiting case Ξ(n,p,q)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Xi (n,p,q) = 0$$\end{document}, we show parabolic C1,Log-Lip\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1, \text {Log-Lip}}$$\end{document} regularity estimates provided F has “better” a priori estimates.