Continuity and minimization of spectrum related with the periodic Camassa-Holm equation

An important point in looking for period solutions of the Camassa-Holm equation is to understand the associated spectral problemy″=14y+λm(t)y. The first aim of this paper is to study the dependence of eigenvalues for the periodic Camassa-Holm Equation on potentials as an infinitely dimensional parameter. To be precise, we prove that as nonlinear functionals of potentials, eigenvalues for the periodic Camassa-Holm Equation are continuous in potentials with respect to the weak topologies in the Lp Lebesgue spaces. The second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic Camassa-Holm Equation when the L1 norm of potentials are given. In order to make our results more applicable, we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations.