EXTREMAL EIGENVALUE PROBLEMS FOR COMPOSITE MEMBRANES .2.

Given an open bounded connected set Ω ⊂RN and a prescribed amount of two homogeneous materials of different density, for small k we characterize the distribution of the two materials in Ω that extremizes the kth eigenvalue of the resulting clamped membrane. We show that these extremizers vary continuously with the proportion of the two constituents. The characterization of the extremizers in terms of level sets of associated eigenfunctions provides geometric information on their respective interfaces. Each of these results generalizes to N dimensions the now classical one-dimensional work of M. G. Krein. © 1990 Springer-Verlag New York Inc.