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Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting. (English) Zbl 1416.35034
The paper under review deals with the investigation of the rate of convergence in periodic homogenization of Hamilton-Jacobi equations. The authors consider the solutions $$u^\epsilon$$ of the problem $\begin{cases} u^\epsilon_t +H(x/\epsilon,Du^\epsilon)=0 & \mathrm{in}\quad \mathbb{R}^n\times (0,+\infty),\\ u^\epsilon(x,0)=g(x) & \mathrm{on}\quad \mathbb{R}^n, \end{cases}$ where the Hamiltonian $$H=H(y,p)$$ is coercive and convex in the $$p$$ variable and $$\mathbb{Z}^n$$-periodic in the $$y$$ variable, and the initial data $$g$$ is bounded and Lipschitz continuous. The authors obtain estimates on the rate of convergence of $$u^\epsilon$$ as $$\epsilon\to 0$$ to the solution $$u$$ of the homogenized problem $\begin{cases} u +\overline{H}(Du)=0 & \mathrm{in}\quad \mathbb{R}^n\times (0,+\infty),\\ u(x,0)=g(x) & \mathrm{on}\quad \mathbb{R}^n, \end{cases}$ where $$\overline{H}$$ is the effective Hamiltonian.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35F21 Hamilton-Jacobi equations 35B40 Asymptotic behavior of solutions to PDEs
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