Minimization of lowest positive periodic eigenvalue for the Camassa–Holm equation with indefinite potential

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