A near-optimal rate of periodic homogenization for convex Hamilton-Jacobi equations (Q2154988)

The paper under review deals with the rate of convergence of a periodic homogenization problem for convex Hamilton-Jacobi equations. More precisely, the author consider the Hamiltonian \(H : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}\) continuous, periodic in the first variable \(x\), and coercive in the second variable \(p\) uniformly in the first variable \(x\). Then the author considers the unique viscosity solution \(u_\epsilon\) of the initial-value problem \[ \left\{ \begin{array}{ll} D_t u^\epsilon (t,x)+H(x/\epsilon, D_xu^\epsilon(t,x))=0 & \mathrm{in} \ \mathbb{R}_{>0}\times \mathbb{R}^d\, ,\\ u^\epsilon(0,x)=u_0(x )& \mathrm{in}\ \mathbb{R}^d\, , \end{array} \right. \] where \(u_0\) is a continuous function from \(\mathbb{R}^d\) to \(\mathbb{R}\). The author proves an estimate of the type \[ |u^\epsilon(t,x)-\overline{u}(t,x)|\leq C \epsilon \log (C+t\epsilon^{-1})\, , \qquad \forall t >0\, , x \in \mathbb{R}^d\, , \] where \(\overline{u}\) is the solution of the effective problem.