We study a one dimensional Dirichlet problem of fourth order and a corresponding "buckling eigenvalue problem" under Dirichlet boundary conditions. These problems may serve as model problems for "Orr-Sommerfeld" like boundary and eigenvalue problems. It turns out that eigenvalue curves in appropriate parameter domains look completely different than for the same equation under so called Navier boundary conditions. Further emphasis is laid on positivity properties, and also here, fundamental differences with Navier conditions arise: It may e.g. happen that one has infinitely many linearly independent positive eigenfunctions. Connections and analogies with the clamped plate boundary value problem on families of deformed domains are discussed.
"Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem." Adv. Differential Equations 7 (2) 177 - 196, 2002.