September  2018, 8(3&4): 855-877. doi: 10.3934/mcrf.2018038

Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors

1. 

School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China

2. 

School of Mathematical Sciences, and SCMS, Fudan University, Shanghai 200433, China

Dedicated to Professor Jiongmin Yong on the Occasion of His 60th Birthday

Received  August 2017 Revised  February 2018 Published  September 2018

Fund Project: This work was supported in part by NSFC Grant 11771097.

Higher eigenvalues of composite materials for anisotropic conductors are considered. To get the existence result for minimizing problems, relaxed problems are introduced by the homogenization method. Then, necessary conditions for minimizers are yielded. Based on the necessary conditions, it is shown that in some cases, optimal conductivities of relaxed minimizing problems can be replaced equivalently by a weighted harmonic mean of conductivities.

Citation: Hongwei Lou, Xueyuan Yin. Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors. Mathematical Control & Related Fields, 2018, 8 (3&4) : 855-877. doi: 10.3934/mcrf.2018038
References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[2]

G. AllaireS. Aubry and F. Jouve, Eigenfrequency optimization in optimal design, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3565-3579.  doi: 10.1016/S0045-7825(00)00284-X.  Google Scholar

[3]

J. Casado-Díaz, Smoothness properties for the optimal mixture of two isotropic materials: the compliance and eigenvalue problems, SIAM J. Control Optim., 53 (2015), 2319-2349.  doi: 10.1137/140971087.  Google Scholar

[4]

J. Casado-DíazJ. Couce-Calvo and J. D. Martín Gómez, Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Control Optim., 43 (2004), 216-239.  doi: 10.1137/S0363012902411714.  Google Scholar

[5]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N. Y., 1953.  Google Scholar

[6]

S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors, Arch. Rational Mech. Anal., 136 (1996), 101-117.  doi: 10.1007/BF02316974.  Google Scholar

[7]

A. R. Díaz and N. Kikuchi, Solutions to shape and topology eigenvalue optimization problems using a homogenization method, Internat. J. Numer. Methods Engrg., 35 (1992), 1487-1502.  doi: 10.1002/nme.1620350707.  Google Scholar

[8]

G. A. Francfort and F. Murat, Optimal bounds for conduction in two-dimensional, twophase, anisotropic media, in Nonclassical Continuum Mechanics (Durham, 1986), London Math. Soc. Lecture Note Ser., 122, Cambridge Univ. Press, Cambridge, (1987), 197–212. doi: 10.1017/CBO9780511662911.013.  Google Scholar

[9]

Y. Grabovsky, The G-closure of two well-ordered, anisotropic conductors, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 423-432.  doi: 10.1017/S0308210500025816.  Google Scholar

[10]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7706-2.  Google Scholar

[11]

S. Kesavan, Homogenization of elliptic eigenvalue problems. Ⅰ, Appl. Math. Optim., 5 (1979), 153-167.  doi: 10.1007/BF01442551.  Google Scholar

[12]

S. Kesavan, Homogenization of elliptic eigenvalue problems. Ⅱ, Appl. Math. Optim., 5 (1979), 197-216.  doi: 10.1007/BF01442554.  Google Scholar

[13]

B. Li and H. Lou, Cesari-type conditions for semilinear elliptic equation with leading term containing controls, Math. Control Relat. Fields, 1 (2011), 41-59.  doi: 10.3934/mcrf.2011.1.41.  Google Scholar

[14]

B. Li and H. Lou, Optimality conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402.  doi: 10.1007/s00245-011-9160-y.  Google Scholar

[15]

B. LiH. Lou and Y. Xu, Relaxation of optimal control problem governed by semilinear elliptic equation with leading term containing controls, Acta Appl. Math., 130 (2014), 205-236.  doi: 10.1007/s10440-013-9843-2.  Google Scholar

[16]

J. LiuL. CaoN. Yan and J. Cui, Multiscale approach for optimal design in conductivity of composite materials, SIAM J. Numer. Anal., 53 (2015), 1325-1349.  doi: 10.1137/13094904X.  Google Scholar

[17]

H. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM Control Optim. Calc. Var., 17 (2011), 975-994.  doi: 10.1051/cocv/2010034.  Google Scholar

[18]

H. Lou and J. Yong, Optimality conditions for semilinear elliptic equations with leading term containing controls, SIAM J. Control Optim., 48 (2009), 2366-2387.  doi: 10.1137/080740301.  Google Scholar

[19]

H. Lou and J. Yong, Optimization of the principal eigenvalue for elliptic operators, preprint. Google Scholar

[20]

K. A. Lurie and A. V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 21-38.  doi: 10.1017/S0308210500019041.  Google Scholar

[21]

Y. MaedaS. NishiwakiK. IzuiM. YoshimuraK. Matsui and K. Terada, Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes, Internat. J. Numer. Methods Engrg., 67 (2006), 597-628.  doi: 10.1002/nme.1626.  Google Scholar

[22]

F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68.  doi: 10.1007/BF02413475.  Google Scholar

[23]

P. Pedregal, Weak limits in nonlinear conductivity, SIAM J. Math. Anal., 47 (2015), 1154-1168.  doi: 10.1137/140960335.  Google Scholar

[24]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, (Italian) Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571–597; errata, ibid. (3), 22 (1968), 673.  Google Scholar

[25]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136–212.  Google Scholar

[26]

L. Tartar, Estimations fines des coefficients homogénéisés, (French) [Fine estimates of homogenized coefficients], in Ennio De Giorgi colloquium (Paris, 1983), Res. Notes in Math., 125, Pitman, Boston, MA, (1985), 168–187.  Google Scholar

[27]

L. Tartar, An introduction to the homogenization method in optimal design, in Optimal shape design (Tróia, 1998), Lecture Notes in Math., 1740, Springer, Berlin, (2000), 47–156. doi: 10.1007/BFb0106742.  Google Scholar

[28]

M. Vrdoljak, Classical optimal design in two-phase conductivity problems, SIAM J. Control Optim., 54 (2016), 2020-2035.  doi: 10.1137/15M1049749.  Google Scholar

show all references

References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  Google Scholar

[2]

G. AllaireS. Aubry and F. Jouve, Eigenfrequency optimization in optimal design, Comput. Methods Appl. Mech. Engrg., 190 (2001), 3565-3579.  doi: 10.1016/S0045-7825(00)00284-X.  Google Scholar

[3]

J. Casado-Díaz, Smoothness properties for the optimal mixture of two isotropic materials: the compliance and eigenvalue problems, SIAM J. Control Optim., 53 (2015), 2319-2349.  doi: 10.1137/140971087.  Google Scholar

[4]

J. Casado-DíazJ. Couce-Calvo and J. D. Martín Gómez, Optimality conditions for nonconvex multistate control problems in the coefficients, SIAM J. Control Optim., 43 (2004), 216-239.  doi: 10.1137/S0363012902411714.  Google Scholar

[5]

R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N. Y., 1953.  Google Scholar

[6]

S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors, Arch. Rational Mech. Anal., 136 (1996), 101-117.  doi: 10.1007/BF02316974.  Google Scholar

[7]

A. R. Díaz and N. Kikuchi, Solutions to shape and topology eigenvalue optimization problems using a homogenization method, Internat. J. Numer. Methods Engrg., 35 (1992), 1487-1502.  doi: 10.1002/nme.1620350707.  Google Scholar

[8]

G. A. Francfort and F. Murat, Optimal bounds for conduction in two-dimensional, twophase, anisotropic media, in Nonclassical Continuum Mechanics (Durham, 1986), London Math. Soc. Lecture Note Ser., 122, Cambridge Univ. Press, Cambridge, (1987), 197–212. doi: 10.1017/CBO9780511662911.013.  Google Scholar

[9]

Y. Grabovsky, The G-closure of two well-ordered, anisotropic conductors, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 423-432.  doi: 10.1017/S0308210500025816.  Google Scholar

[10]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7706-2.  Google Scholar

[11]

S. Kesavan, Homogenization of elliptic eigenvalue problems. Ⅰ, Appl. Math. Optim., 5 (1979), 153-167.  doi: 10.1007/BF01442551.  Google Scholar

[12]

S. Kesavan, Homogenization of elliptic eigenvalue problems. Ⅱ, Appl. Math. Optim., 5 (1979), 197-216.  doi: 10.1007/BF01442554.  Google Scholar

[13]

B. Li and H. Lou, Cesari-type conditions for semilinear elliptic equation with leading term containing controls, Math. Control Relat. Fields, 1 (2011), 41-59.  doi: 10.3934/mcrf.2011.1.41.  Google Scholar

[14]

B. Li and H. Lou, Optimality conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402.  doi: 10.1007/s00245-011-9160-y.  Google Scholar

[15]

B. LiH. Lou and Y. Xu, Relaxation of optimal control problem governed by semilinear elliptic equation with leading term containing controls, Acta Appl. Math., 130 (2014), 205-236.  doi: 10.1007/s10440-013-9843-2.  Google Scholar

[16]

J. LiuL. CaoN. Yan and J. Cui, Multiscale approach for optimal design in conductivity of composite materials, SIAM J. Numer. Anal., 53 (2015), 1325-1349.  doi: 10.1137/13094904X.  Google Scholar

[17]

H. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM Control Optim. Calc. Var., 17 (2011), 975-994.  doi: 10.1051/cocv/2010034.  Google Scholar

[18]

H. Lou and J. Yong, Optimality conditions for semilinear elliptic equations with leading term containing controls, SIAM J. Control Optim., 48 (2009), 2366-2387.  doi: 10.1137/080740301.  Google Scholar

[19]

H. Lou and J. Yong, Optimization of the principal eigenvalue for elliptic operators, preprint. Google Scholar

[20]

K. A. Lurie and A. V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 21-38.  doi: 10.1017/S0308210500019041.  Google Scholar

[21]

Y. MaedaS. NishiwakiK. IzuiM. YoshimuraK. Matsui and K. Terada, Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes, Internat. J. Numer. Methods Engrg., 67 (2006), 597-628.  doi: 10.1002/nme.1626.  Google Scholar

[22]

F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4), 112 (1977), 49-68.  doi: 10.1007/BF02413475.  Google Scholar

[23]

P. Pedregal, Weak limits in nonlinear conductivity, SIAM J. Math. Anal., 47 (2015), 1154-1168.  doi: 10.1137/140960335.  Google Scholar

[24]

S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, (Italian) Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 571–597; errata, ibid. (3), 22 (1968), 673.  Google Scholar

[25]

L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, (1979), 136–212.  Google Scholar

[26]

L. Tartar, Estimations fines des coefficients homogénéisés, (French) [Fine estimates of homogenized coefficients], in Ennio De Giorgi colloquium (Paris, 1983), Res. Notes in Math., 125, Pitman, Boston, MA, (1985), 168–187.  Google Scholar

[27]

L. Tartar, An introduction to the homogenization method in optimal design, in Optimal shape design (Tróia, 1998), Lecture Notes in Math., 1740, Springer, Berlin, (2000), 47–156. doi: 10.1007/BFb0106742.  Google Scholar

[28]

M. Vrdoljak, Classical optimal design in two-phase conductivity problems, SIAM J. Control Optim., 54 (2016), 2020-2035.  doi: 10.1137/15M1049749.  Google Scholar

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