Lower bounds on the eigenvalue ratio with Robin boundary conditions

Abstract

We study the problem of minimizing the ratio of the first eigenvalues of vibrating string equations subject to the Robin boundary conditions for the class of concave weights. We show that, unlike the Dirichlet, Neumann and mixed boundary conditions, the constant weight is not minimizing for the class of concave weights. In addition, we prove a relation between the eigenvalues and real roots of the first Airy functions and their derivatives.