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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.

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F21 Hamilton-Jacobi equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI arXiv
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