We consider Hamilton–Jacobi equations in one space dimension with Hamiltonians of the form \(H(p,x,\omega ) = G(p) +\beta V(x,\omega )\), 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, \(\beta \) and the law of \(V(\cdot ,\omega )\), and discuss a generalization to even-symmetric triple-well Hamiltonians.

我们考虑一维空间中的Hamilton–Jacobi方程，其形式为\（H（p，x，\ omega）= G（p）+ \ beta V（x，\ omega）\），其中\（V（\ cdot，\ omega）\）是单位振幅的平稳和遍历电势。在2016年Armstrong，Tran和Yu的论文中，对所有连续和强制G均建立了此类方程的均质化。在G是双井函数的额外条件下（即，它恰好具有两个局部最小值），我们给出了一个新的且完全有建设性的均质化证明，从而产生了有效哈密顿量\（\ overline {H} \）的公式。我们使用此公式来提供\（\ overline {H} \）图的高度的完整列表有一块扁平的。我们通过分析示例的基本类别来说明我们的结果，强调一些推论来阐明\（\ overline {H} \）对G，\（\ beta \）和\（V（\ cdot，\ omega ）\），并讨论对偶对称三阱哈密顿量的推广。