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 optimization, eigenvalue optimization, two-phase conductors, low contrast regime, asymptotic analysis
@article{COCV_2014__20_2_362_0,
author = {Laurain, Antoine},
title = {Global minimizer of the ground state for two phase conductors in low contrast regime},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {362--388},
year = {2014},
publisher = {EDP Sciences},
volume = {20},
number = {2},
doi = {10.1051/cocv/2013067},
mrnumber = {3264208},
zbl = {1287.49047},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2013067/}
}
TY - JOUR AU - Laurain, Antoine TI - Global minimizer of the ground state for two phase conductors in low contrast regime JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 362 EP - 388 VL - 20 IS - 2 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2013067/ DO - 10.1051/cocv/2013067 LA - en ID - COCV_2014__20_2_362_0 ER -
%0 Journal Article %A Laurain, Antoine %T Global minimizer of the ground state for two phase conductors in low contrast regime %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 362-388 %V 20 %N 2 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2013067/ %R 10.1051/cocv/2013067 %G en %F COCV_2014__20_2_362_0
Laurain, Antoine. Global minimizer of the ground state for two phase conductors in low contrast regime. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 362-388. doi: 10.1051/cocv/2013067
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