Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17.
Juni 136, 10623
Revised: 24 June 2013
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.
Mathematics Subject Classification: 49Q10 / 35P15 / 49R05 / 47A55 / 34E10
Key words: Shape optimization / eigenvalue optimization / two-phase conductors / low contrast regime / asymptotic analysis
Financial support by the DFG Research Center Matheon “Mathematics for key technologies” through the MATHEON-Project C37 “Shape/Topology optimization methods for inverse problems”.
© EDP Sciences, SMAI, 2014