Extremal values of eigenvalues of Sturm–Liouville operators with potentials in L1 balls

This paper is a continuation of Zhang [M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Ser. A 51 (2008) 1036–1058; M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L1 balls, J. Differential Equations 246 (2009) 4188–4220]. Given a potential q∈Lp([0,1],R), p∈[1,∞]. We use λm(q) to denote the mth Dirichlet eigenvalue of the Sturm–Liouville operator with potential q(t), where m∈N. The minimal value Lm,p(r) and the maximal value Mm,p(r) of λm(q) with potentials q in the Lp ball of radius r are well defined. In this paper, we will exploit the continuity of λm(q) in q with weak topologies and the variational method to give characterizations of Lm,p(r) and Mm,p(r) when p∈(1,∞). By using the limiting approach as p↓1, we find that the most important extremal values Lm,1(r) and Mm,1(r) can be evaluated explicitly using some elementary functions of r. The corresponding extremal problems for Neumann eigenvalues and some periodic eigenvalues will be reduced to Lm,p(r) and Mm,p(r).