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.

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.

DOI : 10.1051/cocv/2013067
Classification : 49Q10, 35P15, 49R05, 47A55, 34E10
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] A. Alvino, G. Trombetti and P.-L. Lions, On optimization problems with prescribed rearrangements. Nonlinear Anal. 13 (1989) 185-220. | MR | Zbl

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

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

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

[5] C. Conca, R. Mahadevan and L. Sanz, 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] S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal. 136 (1996) 101-117. | MR | Zbl

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

[8] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition. | MR | Zbl

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

[10] M.G. Krein, 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] M.G. Krein and M.A. Rutman, 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] G.N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England (1944). | JFM | MR | Zbl

Cité par Sources :