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}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, doi = {10.1051/cocv/2013067}, mrnumber = {3264208}, zbl = {1287.49047}, language = {en}, url = {http://archive.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://archive.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://archive.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, Volume 20 (2014) no. 2, pp. 362-388. doi : 10.1051/cocv/2013067. http://archive.numdam.org/articles/10.1051/cocv/2013067/

[1] On optimization problems with prescribed rearrangements. Nonlinear Anal. 13 (1989) 185-220. | MR | Zbl

, and ,[2] Hardy's inequality and its extensions. Pacific J. Math. 11 (1961) 39-61. | MR | Zbl

,[3] Minimization of the ground state for two phase conductors in low contrast regime. SIAM J. Appl. Math. 72 (2012) 1238-1259. | MR

, and ,[4] An extremal eigenvalue problem for a two-phase conductor in a ball. Appl. Math. Optim. 60 (2009) 173-184. | MR | Zbl

, and ,[5] 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

, and ,[6] Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal. 136 (1996) 101-117. | MR | Zbl

and ,[7] On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Appl. Math. Optim. 63 (2011) 45-74. | MR | Zbl

and ,[8] Inequalities, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition. | MR | Zbl

, and ,[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] Linear operators leaving invariant a cone in a banach space. Amer. Math. Soc. Transl. (1950) 26. | MR | Zbl

and ,[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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