Extremal values of eigenvalues of Sturm–Liouville operators with potentials in [formula omitted] balls

Abstract

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 L 1 balls, J. Differential Equations 246 (2009) 4188–4220]. Given a potential q ∈ L p ( [ 0 , 1 ] , R ) , p ∈ [ 1 , ∞ ] . We use λ m ( q ) to denote the mth Dirichlet eigenvalue of the Sturm–Liouville operator with potential q ( t ) , where m ∈ N . The minimal value L m , p ( r ) and the maximal value M m , p ( r ) of λ m ( q ) with potentials q in the L p ball of radius r are well defined. In this paper, we will exploit the continuity of λ m ( q ) in q with weak topologies and the variational method to give characterizations of L m , p ( r ) and M m , p ( r ) when p ∈ ( 1 , ∞ ) . By using the limiting approach as p ↓ 1 , we find that the most important extremal values L m , 1 ( r ) and M m , 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 L m , p ( r ) and M m , p ( r ) .