Identification of composite membranes for extremal eigenvalue problems

Given an open bounded connected set Ω⊂R^N and a prescribed amount of two homogeneous materials of different density, for small k the authors characterize the distribution of the two materials in Ω that extremizes the kth eigenvalue of the resulting clamped membrane. It is shown 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 the respective interfaces. Each of these results generalizes to N dimensions the one-dimensional work of M.G. Krein (1955). In addition to providing a first attack on the analytical study of the vibration of composites, this work has relevance in those fields of medicine and biology where composite membranes abound