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Minimization of the ground state of the mixture of two conducting materials in a small contrast regime
Author(s) -
Conca Carlos,
Dambrine Marc,
Mahadevan Rajesh,
Quintero Duver
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3797
Subject(s) - eigenvalues and eigenvectors , contrast (vision) , operator (biology) , minification , mathematics , relaxation (psychology) , asymptotic expansion , dirichlet distribution , ground state , domain (mathematical analysis) , mathematical analysis , order (exchange) , mathematical optimization , physics , boundary value problem , quantum mechanics , chemistry , social psychology , biochemistry , finance , repressor , transcription factor , optics , economics , gene , psychology
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.