Minimization of lowest positive periodic eigenvalue for the Camassa–Holm equation with indefinite potential
Volume 268 / 2023
Abstract
Given a measure $\mu \in \mathcal M_{\rm sgn},$ we study the periodic eigenvalues of the measure differential equation $${\rm d}y^{\bullet }= \tfrac {1}{4} y\,{\rm d}t + \lambda y \,{\rm d}\mu (t).$$ We present a variational characterization of the lowest positive periodic eigenvalues and prove a strong continuous dependence of eigenvalues on potentials as an infinite-dimensional parameter. The optimal lower bound of the lowest positive eigenvalues is also obtained when the total variation of potentials is given. Our main results can be directly applied to the periodic spectrum of the Camassa–Holm equation $$y”=\tfrac {1}{4}y +\lambda m(t)y.$$ In particular, we obtain the optimal lower bound for the lowest positive periodic eigenvalues by allowing the potential $m$ to change sign.
