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Global minimizer of the ground state for two phase conductors in low contrast regime∗∗∗

Published online by Cambridge University Press:  03 March 2014

Antoine Laurain
Antoine Laurain*
Affiliation:
Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17. Juni 136, 10623 Berlin, Germany. laurain@math.tu-berlin.de
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Abstract

The problem of distributing two conducting materials with a prescribed volume ratio in a ball so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions is considered in two and three dimensions. The gap ε between the two conductivities is assumed to be small (low contrast regime). The main result of the paper is to show, using asymptotic expansions with respect to ε and to small geometric perturbations of the optimal shape, that the global minimum of the first eigenvalue in low contrast regime is either a centered ball or the union of a centered ball and of a centered ring touching the boundary, depending on the prescribed volume ratio between the two materials.

Keywords

Shape optimizationeigenvalue optimizationtwo-phase conductorslow contrast regimeasymptotic analysis
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 362 - 388
Copyright
© EDP Sciences, SMAI, 2014

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