Minimizations of positive periodic and Dirichlet eigenvalues for general indefinite Sturm-Liouville problems (Q6049890)

Consider the indefinite Sturm-Liouville problem \(y'' = q(t) y + \lambda m(t) y\) on \([0,1]\) with continuous potentials \(q, \, m\) such that \(q(t) \geq \alpha\) with a given constant \(\alpha \geq 0\) and \(m\) changes its sign on \([0,1]\). Due to this sign change it is clear that in case of periodic, anti-periodic or Dirichlet boundary conditions the eigenvalues form a ``double infinite'' real sequence accumulating at \(\infty\) and \(-\infty\). The main focus of the paper is on a sharp estimate for the lowest positive periodic eigenvalue \(\lambda_0^+(q,m)\). More precisely, an explicit formula is presented for the infimum of all such values \(\lambda_0^+(q,m)\) where \(r_1 = \int_0^1 |q(t)| \, dt, \; r_2 = \int_0^1 |m(t)| \, dt\) and \(h = \int_0^1 m(t) \, dt\) are fixed given numbers. Here, different approaches are required for the two cases \(\alpha > 0\) and \(\alpha = 0\) and different results are found. Furthermore, similar estimates are obtained for all (not only the lowest) positive Dirichlet eigenvalues. All these estimates are at first formulated in the more general setting of the measure differential equation \(d y^{\bullet} = y(t) d\mu(t) + \lambda y(t) d\nu(t)\) where \(\mu\) and \(\nu\) are certain measures. In a first step it is observed that the associated lowest positive periodic eigenvalue \(\lambda_0^+(\mu, \nu)\) depends coninuously on \(\mu\) and \(\nu\) in a suitable sense. An application of the results to the Camassa-Holm equation (where \(q(t) = \frac{1}{4}\)) leads to an estimate for \(\lambda_0^+(q,m)\) which is already known from [\textit{J. Chu} and \textit{G. Meng}, Stud. Math. 268, No. 3, 241--258 (2023; Zbl 1518.34024)].