Modeling and justification of eigenvalue problems for junctions between elastic structures

We consider a problem in three-dimensional linearized elasticity, posed over a domain consisting of a plate with thickness 2ε inserted into a solid whose Lamé constants and density are independent of ε. If the Lamé constants of the material constituting the plate vary as ε^-3 and its density as ε^-1, we show that the solutions of the three-dimensional eigenvalue problem converge, as ε approaches zero, to the solutions of a "coupled," "pluri-dimensional" eigenvalue problem of a new type, posed simultaneously over a three-dimensional open set with a slit and a two-dimensional open set. © 1989.