Abstract
Given an open bounded connected set Ω ⊂R N and a prescribed amount of two homogeneous materials of different density, for smallk we characterize the distribution of the two materials in Ω that extremizes thekth 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 toN dimensions the now classical one-dimensional work of M. G. Krein.
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Communicated by D. Kinderlehrer
The work of the first author was supported in part by NSF Grant DMS-8201719 (A. Manitius, P. I.), an IBM fellowship, a GE teaching incentive, and DARPA Contract F49620-87-C-0065. That of the second author was supported in part by ONR Grant N00014-84-5-516, AFOSR Grant AFOSR-86-0180, and NSF Grant DMS-8713722.
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Cox, S.J., McLaughlin, J.R. Extremal eigenvalue problems for composite membranes, II. Appl Math Optim 22, 169–187 (1990). https://doi.org/10.1007/BF01447326
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DOI: https://doi.org/10.1007/BF01447326