A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem y″ = 1 4y + λm(t)y. We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the L1 norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure differential equations to make our results more applicable. Second, Based on the relationship between the minimization problem of the smallest eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation.

中文翻译：

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Camassa-Holm 方程的特征值的最小化

寻求 Camassa-Holm 方程解的关键基础是理解相关的谱问题 是的” = 1 4是的 + λ米(吨)是的. 我们将在本文中研究具有 Neumann 边界条件的 Camassa-Holm 方程的最小特征值的最优下界，当大号1给出了电位范数。首先，我们将研究测度微分方程中最小特征值的最优下界，以使我们的结果更适用。其次，基于ODE最小特征值最小化问题与MDE最小化问题之间的关系，我们找到了Camassa-Holm方程最小特征值的显式最优下界。