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arXiv · 1808.06129

Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension

Son N. T. Tu
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Abstract

Let $u^\varepsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n = 1$.

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Son N. T. Tu. 2018-08-18. Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension. https://doi.org/10.3233/asy-201599

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