Extremal norms of the potentials recovered from inverse Dirichlet problems

Consider the Sturm-Liouville eigenvalue problem , where , and its spectrum is denoted by . For a real number λ, define and . We will set up a formula for explicitly in terms of λ and specify where the infimum can be attained. As an application, we will give the extremal values of the nth eigenvalue of the Dirichlet problem for potentials on a sphere , . The proofs are based on a new Lyapunov-type inequality for Sturm-Liouville equations with potentials.