Minimization of the ground state of the mixture of two conducting materials in a small contrast regime. (22nd February 2016)

• Record Type: Journal Article
• Authors: Conca, Carlos; Dambrine, Marc; Mahadevan, Rajesh; Quintero, Duver
• Is Part Of: Mathematical methods in the applied sciences. Volume 39:Number 13(2016:Sep. 15)
• Page Start: 3549
• Page End: 3564
• Publication Date: 2016-02-22
• Subjects: shape optimization -- eigenvalue problem -- homogenization; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3797
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second‐order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second‐order method. Copyright © 2016 John Wiley & Sons, Ltd.