×

Minimization of the first nonzero eigenvalue problem for two-phase conductors with Neumann boundary conditions. (English) Zbl 1443.49053

Summary: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed on circles, ellipses, annuli, and L-shaped domains.

MSC:

49R05 Variational methods for eigenvalues of operators
49J20 Existence theories for optimal control problems involving partial differential equations
35P15 Estimates of eigenvalues in context of PDEs
47J30 Variational methods involving nonlinear operators
47A55 Perturbation theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Alvino, G. Trombetti, and P. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal., 13 (1989), pp. 185-220. · Zbl 0678.49003
[2] C. Anedda, F. Cuccu, and G. Porru, Minimization of the first eigenvalue in problems involving the bi-Laplacian, Rev. Mat. Teor. Apl., 16 (2009), pp. 127-136.
[3] D. O. Banks, Bounds for the eigenvalues of nonhomogeneous hinged vibrating rods, J. Math. Mech., 16 (1967), pp. 949-966. · Zbl 0189.26003
[4] P. R. Beesack, Isoperimetric inequalities for the nonhomogeneous clamped rod and plate, J. Math. Mech, 8 (1959), pp. 471-482. · Zbl 0086.38401
[5] L. Cadeddu, M. A. Farina, and G. Porru, Optimization of the principal eigenvalue under mixed boundary conditions, Electron. J. Differential Equations, 2014 (2014), pp. 1-17. · Zbl 1310.47024
[6] J. Casado-Díaz, A characterization result for the existence of a two-phase material minimizing the first eigenvalue, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), pp. 1215-1226. · Zbl 1379.49044
[7] S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi, Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Comm. Math. Phys., 214 (2000), pp. 315-337, https://doi.org/10.1007/PL00005534. · Zbl 0972.49030
[8] W. Chen, C.-S. Chou, and C.-Y. Kao, Minimizing eigenvalues for inhomogeneous rods and plates, J. Sci. Comput., 69 (2016), pp. 983-1013. · Zbl 1397.74197
[9] W. Chen, K. Diest, C.-Y. Kao, D. E. Marthaler, L. A. Sweatlock, and S. Osher, Gradient based optimization methods for metamaterial design, in Numerical Methods for Metamaterial Design, Springer, 2013, pp. 175-204.
[10] M. Chugunova, B. Jadamba, C.-Y. Kao, C. Klymko, E. Thomas, and B. Zhao, Study of a mixed dispersal population dynamics model, in Topics in Numerical Partial Differential Equations and Scientific Computing, Springer, 2016, pp. 51-77. · Zbl 1384.92049
[11] C. Conca, M. Dambrine, R. Mahadevan, and D. Quintero, Minimization of the ground state of the mixture of two conducting materials in a small contrast regime, Math. Methods Appl. Sci., 39 (2016), pp. 3549-3564. · Zbl 1343.15007
[12] 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), pp. 1238-1259, https://doi.org/10.1137/110847822. · Zbl 1321.49078
[13] C. Conca, R. Mahadevan, and L. Sanz, An extremal eigenvalue problem for a two-phase conductor in a ball, Appl. Math. Optim., 60 (2009), pp. 173-184. · Zbl 1179.49052
[14] C. Conca, R. Mahadevan, and L. Sanz, Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball, ESAIM Proc., 27 (2009), pp. 311-321. · Zbl 1167.49038
[15] S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors, Arch. Rational Mech. Anal., 136 (1996), pp. 101-117. · Zbl 0914.49011
[16] S. J. Cox, The two phase drum with the deepest bass note, Japan J. Indust. Appl. Math., 8 (1991), pp. 345-355, https://doi.org/10.1007/BF03167141. · Zbl 0755.35029
[17] S. J. Cox and D. C. Dobson, Band structure optimization of two-dimensional photonic crystals in h-polarization, Journal of Computational Physics, 158 (2000), pp. 214-224. · Zbl 0949.65115
[18] S. J. Cox and D. C. Dobson, Maximizing band gaps in two-dimensional photonic crystals, SIAM J. Appl. Math., 59 (1999), pp. 2108-2120, https://doi.org/10.1137/S0036139998338455. · Zbl 1027.78521
[19] F. Cuccu, B. Emamizadeh, and G. Porru, Optimization of the first eigenvalue in problems involving the bi-Laplacian, Proc. Amer. Math. Soc., 137 (2009), pp. 1677-1687. · Zbl 1163.35025
[20] F. Cuccu and G. Porru, Maximization of the first eigenvalue in problems involving the bi-Laplacian, Nonlinear Anal., 71 (2009), pp. e800-e809. · Zbl 1238.35066
[21] M. Dambrine and D. Kateb, On the shape sensitivity of the first Dirichlet eigenvalue for two-phase problems, Appl. Math. Optim., 63 (2011), pp. 45-74. · Zbl 1207.49055
[22] L. He, C.-Y. Kao, and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods, J. Comput. Phys., 225 (2007), pp. 891-909. · Zbl 1122.65057
[23] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Springer Science & Business Media, 2006. · Zbl 1109.35081
[24] M. Hintermüller, C.-Y. Kao, and A. Laurain, Principal eigenvalue minimization for an elliptic problem with indefinite weight and Robin boundary conditions, Appl. Math. Optim., 65 (2012), pp. 111-146. · Zbl 1242.49094
[25] D. Kang and C.-Y. Kao, Minimization of inhomogeneous biharmonic eigenvalue problems, Appl. Math. Model., 51 (2017), pp. 587-604. · Zbl 1480.49046
[26] C.-Y. Kao, Y. Lou, and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Engrg., 5 (2008), pp. 315-335. · Zbl 1167.35426
[27] C.-Y. Kao, S. Osher, and E. Yablonovitch, Maximizing band gaps in two-dimensional photonic crystals by using level set methods, Appl. Phys. B, 81 (2005), pp. 235-244.
[28] C.-Y. Kao and F. Santosa, Maximization of the quality factor of an optical resonator, Wave Motion, 45 (2008), pp. 412-427. · Zbl 1231.78007
[29] C.-Y. Kao and S. Su, Efficient rearrangement algorithms for shape optimization on elliptic eigenvalue problems, J. Sci. Comput., 54 (2013), pp. 492-512. · Zbl 1263.65107
[30] I. M. Karabash, Nonlinear eigenvalue problem for optimal resonances in optical cavities, Math. Model. Nat. Phenom., 8 (2013), pp. 143-155. · Zbl 1259.78050
[31] 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. (2), 1 (1955), pp. 163-187.
[32] J. Lamboley, A. Laurain, G. Nadin, and Y. Privat, Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions, Calc. Var. Partial Differential Equations, 55 (2016), 144. · Zbl 1366.49004
[33] A. Laurain, Global minimizer of the ground state for two phase conductors in low contrast regime, ESAIM Control Optim. Calc. Var., 20 (2014), pp. 362-388. · Zbl 1287.49047
[34] J. Lin and F. Santosa, Resonances of a finite one-dimensional photonic crystal with a defect, SIAM J. Appl. Math., 73 (2013), pp. 1002-1019, https://doi.org/10.1137/120897304. · Zbl 1293.35316
[35] Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), pp. 275-292. · Zbl 1185.35059
[36] M. Maksimović, M. Hammer, and E. B. van Groesen, Coupled optical defect microcavities in one-dimensional photonic crystals and quasi-normal modes, Opt. Engrg., 47 (2008), pp. 114601-114601.
[37] K. Matsue and H. Naito, Numerical studies of the optimization of the first eigenvalue of the heat diffusion in inhomogeneous media, Japan J. Indust. Appl. Math., 32 (2015), pp. 489-512. · Zbl 1323.35187
[38] H. Men, K. Y. Lee, R. M. Freund, J. Peraire, and S. G. Johnson, Robust topology optimization of three-dimensional photonic-crystal band-gap structures, Opt. Express, 22 (2014), pp. 22632-22648.
[39] H. Men, N.-C. Nguyen, R. M. Freund, K.-M. Lim, P. A. Parrilo, and J. Peraire, Design of photonic crystals with multiple and combined band gaps, Phys. Rev. E, 83 (2011), 046703.
[40] F. Meng, B. Jia, and X. Huang, Topology-optimized 3D photonic structures with maximal omnidirectional bandgaps, Adv. Theory Simul., 1 (2018), 1800122.
[41] A. Mohammadi and M. Yousefnezhad, Optimal ground state energy of two-phase conductors, Electron. J. Differential Equations, 171 (2014), pp. 1-7. · Zbl 1296.49044
[42] S. J. Osher and F. Santosa, Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171 (2001), pp. 272-288. · Zbl 1056.74061
[43] B. Osting, Bragg structure and the first spectral gap, Appl. Math. Lett., 25 (2012), pp. 1926-1930. · Zbl 1252.35175
[44] B. Schwarz, Some results on the frequencies of nonhomogeneous rods, J. Math. Anal. Appl., 5 (1962), pp. 169-175. · Zbl 0107.19501
[45] O. Sigmund and K. Hougaard, Geometric properties of optimal photonic crystals, Phys. Rev. Lett., 100 (2008), 153904.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.