Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media

  • Kaname MatsueEmail author
  • Hisashi Naito
Original Paper Area 1


In this paper, we study optimization of the first eigenvalue of \(-\nabla \cdot (\rho (x) \nabla u) = \lambda u\) in a bounded domain \(\Omega \subset {\mathbb {R}}^n\) under several constraints for the function \(\rho \). We consider this problem in various boundary conditions and various topologies of domains. As a result, we numerically observe several common criteria for \(\rho \) for optimizing eigenvalues in terms of corresponding eigenfunctions, which are independent of topology of domains and boundary conditions. Geometric characterizations of optimizers are also numerically observed.


Eigenvalue problem Topology optimization Dependence of optimizers on topology 

Mathematics Subject Classification

35Q93 49J20 65N25 74G15 



This research was partially supported by JST, CREST: A Mathematical Challenge to a New Phase of Material Science, Based on Discrete Geometric Analysis. KM was partially supported by Coop with Math Program, a commissioned project by MEXT. HN was partially supported by Grants-in-Aid for Scientific Research (C) (No. 26400067). We thank Prof. Hideyuki Azegami for providing us with very meaningful advice for our computational study. We also thank Prof. Motoko Kotani for introducing us to this study and giving us a lot of advice for writing this paper.


  1. 1.
    Bandle, C.: Isoperimetric inequalities and applications. In: Monographs and Studies in Mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston, Mass, London (1980)Google Scholar
  2. 2.
    Chanillo, S., Grieser, D., Imai, M., Kurata, K., Ohnishi, I.: Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Commun. Math. Phys. 214, 315–337 (2000). doi: 10.1007/PL00005534
  3. 3.
    Conca, C., Laurain, A., Mahadevan, R.: Minimization of the ground state for two phase conductors in low contrast regime. SIAM J. Appl. Math. 72, 1238–1259 (2012). doi: 10.1137/110847822 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Conca, C., Laurain, A., Mahadevan, R., Quintero, D.: Minimization of the ground state of the mixture of two conducting materials in a small contrast regime (2014). arXiv:1408.4981
  5. 5.
    Conca, C., Mahadevan, R., Sanz, L.: Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball. In: CANUM 2008, ESAIM Proc., vol. 27, pp. 311–321. EDP Sci., Les Ulis (2009). doi: 10.1051/proc/2009029
  6. 6.
    Courant, R., Hilbert, D.: Methods of mathematical physics, vols. I and II. Interscience Publishers, New York-London (1962)Google Scholar
  7. 7.
    Cox, S., Lipton, R.: Extremal eigenvalue problems for two-phase conductors. Arch. Ration. Mech. Anal. 136, 101–117 (1996). doi: 10.1007/BF02316974 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cox, S.J., McLaughlin, J.R.: Extremal eigenvalue problems for composite membranes. I. Appl. Math. Optim. 22, 153–167 (1990). doi: 10.1007/BF01447325 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cox, S.J., McLaughlin, J.R.: Extremal eigenvalue problems for composite membranes, II. Appl. Math. Optim. 22, 169–187 (1990). doi: 10.1007/BF01447326 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Crandall, M.G., Lions, P.-L.: Viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983). doi: 10.2307/1999343 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hecht, F.: New development in freefem\(++\). J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Krein, M.G.: On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Am. Math. Soc. Transl. (2) 1, 163–187 (1955)MathSciNetGoogle Scholar
  13. 13.
    Lou, Y., Yanagida, E.: Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics. Jpn. J. Ind. Appl. Math. 23, 275–292 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988). doi: 10.1016/0021-9991(88)90002-2 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Osher, S.J., Santosa, F.: Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171, 272–288 (2001). doi: 10.1006/jcph.2001.6789 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Trudinger, N.S.: Maximum principles for linear, non-uniformly elliptic operators with measurable coefficients. Math. Z. 156, 291–301 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.The Institute of Statistical MathematicsTachikawaJapan
  2. 2.Graduate school of MathematicsNagoya UniversityNagoyaJapan

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