Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${D^2 S^\varepsilon(t,x)\leq g(t)I}$$\end{document} for the solution of the viscous Hamilton–Jacobi equation\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}$$\end{document}The smallest possible positive function g(t) is explicitly given in terms of the semiconvexity and semiconcavity parameters of V and S0, respectively. The optimal g does not depend on the viscosity parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon >0 }$$\end{document} . The potential V and the initial function S0 are allowed to grow quadratically.