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.
Mots-clés : 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},
publisher = {EDP-Sciences},
volume = {20},
number = {2},
year = {2014},
doi = {10.1051/cocv/2013067},
mrnumber = {3264208},
zbl = {1287.49047},
language = {en},
url = {http://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 - http://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 http://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. http://www.numdam.org/articles/10.1051/cocv/2013067/
[1] , and , On optimization problems with prescribed rearrangements. Nonlinear Anal. 13 (1989) 185-220. | MR | Zbl
[2] , Hardy's inequality and its extensions. Pacific J. Math. 11 (1961) 39-61. | MR | Zbl
[3] , and , Minimization of the ground state for two phase conductors in low contrast regime. SIAM J. Appl. Math. 72 (2012) 1238-1259. | MR
[4] , and , An extremal eigenvalue problem for a two-phase conductor in a ball. Appl. Math. Optim. 60 (2009) 173-184. | MR | Zbl
[5] , and , Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball, in vol. 27 of CANUM 2008, ESAIM Proc. EDP Sciences, Les Ulis (2009) 311-321 | MR | Zbl
[6] and , Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal. 136 (1996) 101-117. | MR | Zbl
[7] and , On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Appl. Math. Optim. 63 (2011) 45-74. | MR | Zbl
[8] , and , Inequalities, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition. | MR | Zbl
[9] , Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR | Zbl
[10] , On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Amer. Math. Soc. Transl. 1 (1955) 163-187. | MR | Zbl
[11] and , Linear operators leaving invariant a cone in a banach space. Amer. Math. Soc. Transl. (1950) 26. | MR | Zbl
[12] F Rellich, Perturbation Theory of Eigenvalue Problems, Notes on mathematics and its applications. Gordon and Breach, New York (1969). | MR | Zbl
[13] , A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England (1944). | JFM | MR | Zbl
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