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Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case | | Atilla Yilmaz
; | | Date: |
15 Jul 2020 | | Abstract: | We consider Hamilton-Jacobi equations in one space dimension with
Hamiltonians of the form $H(p,x,omega) = G(p) + eta V(x,omega)$, where
$V(cdot,omega)$ is a stationary and ergodic potential of unit amplitude. The
homogenization of such equations is established in a 2016 paper of Armstrong,
Tran and Yu for all continuous and coercive $G$. Under the extra condition that
$G$ is a double-well function (i.e., it has precisely two local minima), we
give a new and fully constructive proof of homogenization which yields a
formula for the effective Hamiltonian $overline H$. We use this formula to
provide a complete list of the heights at which the graph of $overline H$ has
a flat piece. We illustrate our results by analyzing basic classes of examples,
highlight some corollaries that clarify the dependence of $overline H$ on $G$,
$eta$ and the law of $V(cdot,omega)$, and discuss a generalization to
even-symmetric triple-well Hamiltonians. | | Source: | arXiv, 2007.7854 | | Services: | Forum | Review | PDF | Favorites | |
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