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

Volume 268 / 2023

Jifeng Chu, Gang Meng Studia Mathematica 268 (2023), 241-258 MSC: Primary 34L15; Secondary 34L40, 76B15. DOI: 10.4064/sm211019-20-6 Published online: 28 September 2022


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.


  • Jifeng ChuDepartment of Mathematics
    Shanghai Normal University
    Shanghai 200234, China
  • Gang MengSchool of Mathematical Sciences
    University of Chinese Academy of Sciences
    Beijing 100049, China

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