Extremal norms of the potentials recovered from inverse Dirichlet problems

Consider the Sturm-Liouville eigenvalue problem -y"(x) + q(x) y(x) = lambda y(x), x is an element of [0,1], y(0)= y(1) = 0, where q is an element of L-1[0, 1], and its spectrum is denoted by sigma(q). For a real number lambda, define Omega(lambda)= {q is an element of L-1[0, 1]: lambda is an element of sigma(q)} and E(lambda)= inf{parallel to q parallel to: q is an element of Omega(lambda)}. We will set up a formula for E (lambda) explicitly in terms of. 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 L-1[0, 1], n >=[1. The proofs are based on a new Lyapunov-type inequality for Sturm-Liouville equations with potentials.