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.

Classification: 49Q10, 35P15, 49R05, 47A55, 34E10

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}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, pages = {362-388}, doi = {10.1051/cocv/2013067}, zbl = {1287.49047}, mrnumber = {3264208}, language = {en}, url = {http://www.numdam.org/item/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://www.numdam.org/item/COCV_2014__20_2_362_0/

[1] On optimization problems with prescribed rearrangements. Nonlinear Anal. 13 (1989) 185-220. | MR 979040 | Zbl 0678.49003

, and ,[2] Hardy's inequality and its extensions. Pacific J. Math. 11 (1961) 39-61. | MR 121449 | Zbl 0103.03503

,[3] Minimization of the ground state for two phase conductors in low contrast regime. SIAM J. Appl. Math. 72 (2012) 1238-1259. | MR 2968770 | Zbl pre06111118

, and ,[4] An extremal eigenvalue problem for a two-phase conductor in a ball. Appl. Math. Optim. 60 (2009) 173-184. | MR 2524685 | Zbl 1179.49052

, 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 2562647 | Zbl 1167.49038

, and ,[6] Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal. 136 (1996) 101-117. | MR 1423004 | Zbl 0914.49011

and ,[7] On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems. Appl. Math. Optim. 63 (2011) 45-74. | MR 2746730 | Zbl 1207.49055

and ,[8] Inequalities, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition. | MR 944909 | Zbl 0634.26008

, and ,[9] Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR 2251558 | Zbl 1109.35081

,[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 73776 | Zbl 0066.33404

,[11] Linear operators leaving invariant a cone in a banach space. Amer. Math. Soc. Transl. (1950) 26. | MR 38008 | Zbl 0030.12902

and ,[12] F Rellich, Perturbation Theory of Eigenvalue Problems, Notes on mathematics and its applications. Gordon and Breach, New York (1969). | MR 240668 | Zbl 0181.42002

[13] A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England (1944). | JFM 48.0412.02 | MR 10746 | Zbl 0174.36202

,