Optimal estimates of the first gap for periodic Sturm-Liouville problems with indefinite weights

The present paper studies the first gap of periodic Sturm-Liouville problems with indefinite weights which arise from Camassa-Holm equations. The optimal infimum of the first gap is obtained and the optimal indefinite weight is investigated. The main results are based on the trace formula and the generalized Lyapunov inequalities for periodic Sturm-Liouville problems with indefinite weights. Moreover, we give a confirm answer to an open problem given in [9] .