Global minimizer of the ground state for two phase conductors in low contrast regime

Antoine Laurain

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 362-388
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Laurain, Antoine. "Global minimizer of the ground state for two phase conductors in low contrast regime." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 362-388. <http://eudml.org/doc/272805>.

@article{Laurain2014,
abstract = {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.},
author = {Laurain, Antoine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {shape optimization; eigenvalue optimization; two-phase conductors; low contrast regime; asymptotic analysis},
language = {eng},
number = {2},
pages = {362-388},
publisher = {EDP-Sciences},
title = {Global minimizer of the ground state for two phase conductors in low contrast regime},
url = {http://eudml.org/doc/272805},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 2
SP - 362
EP - 388
AB - 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.
LA - eng
KW - shape optimization; eigenvalue optimization; two-phase conductors; low contrast regime; asymptotic analysis
UR - http://eudml.org/doc/272805
ER -

References

top
  1. [1] A. Alvino, G. Trombetti and P.-L. Lions, On optimization problems with prescribed rearrangements. Nonlinear Anal.13 (1989) 185–220. Zbl0678.49003MR979040
  2. [2] P.R. Beesack, Hardy’s inequality and its extensions. Pacific J. Math.11 (1961) 39–61. Zbl0103.03503MR121449
  3. [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. Zbl1321.49078MR2968770
  4. [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. Zbl1179.49052MR2524685
  5. [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 Zbl1167.49038MR2562647
  6. [6] S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors. Arch. Rational Mech. Anal.136 (1996) 101–117. Zbl0914.49011MR1423004
  7. [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. Zbl1207.49055MR2746730
  8. [8] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edition. Zbl0634.26008MR944909
  9. [9] A. Henrot, Extremum problems for eigenvalues of elliptic operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). Zbl1109.35081MR2251558
  10. [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. Zbl0066.33404MR73776
  11. [11] M.G. Krein and M.A. Rutman, Linear operators leaving invariant a cone in a banach space. Amer. Math. Soc. Transl. (1950) 26. Zbl0030.12902MR38008
  12. [12] F Rellich, Perturbation Theory of Eigenvalue Problems, Notes on mathematics and its applications. Gordon and Breach, New York (1969). Zbl0181.42002MR240668
  13. [13] G.N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England (1944). Zbl0174.36202MR10746JFM48.0412.02

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.