# American Institute of Mathematical Sciences

April  2018, 38(4): 2079-2092. doi: 10.3934/dcds.2018085

## Minimization of the lowest eigenvalue for a vibrating beam

 1 LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China 3 LMIB and School of Mathematics and Systems Science, Beihang University, Beijing, China

* Corresponding author: Zhikun She; Email: zhikun.she@buaa.edu.cn

Received  May 2017 Published  January 2018

Fund Project: The third author is supported by the National Natural Science Foundation of China (Grant No. 11671378) and the Fund of UCAS. The fourth author is supported by the National Natural Science Foundation of China (Grants No. 11371047 and No. 11422111).

In this paper we solve the minimization problem of the lowest eigenvalue for a vibrating beam. Firstly, based on the variational method, we establish the basic theory of the lowest eigenvalue for the fourth order measure differential equation (MDE). Secondly, we build the relationship between the minimization problem of the lowest eigenvalue for the ODE and the one for the MDE. Finally, with the help of this built relationship, we find the explicit optimal bound of the lowest eigenvalue for a vibrating beam.

Citation: Quanyi Liang, Kairong Liu, Gang Meng, Zhikun She. Minimization of the lowest eigenvalue for a vibrating beam. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2079-2092. doi: 10.3934/dcds.2018085
##### References:
 [1] M. Carter and B. van Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction Springer-Verlag, New York, 2000. [2] R. Courant and D. Hilbert, Methods of Mathematical Physics Wiley, New York, 1953. [3] Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ., 16 (2009), 299-313. [4] X. Jiang, K. Liu, G. Meng and Z. She, Continuity of the eigenvalues for a vibrating beam, Appl. Math. Lett., 67 (2017), 60-66.  doi: 10.1016/j.aml.2016.12.006. [5] R. E. Megginson, An Introduction to Banach Space Theory Graduate Texts in Mathematics, 183 Springer-Verlag, New York, 1998. [6] G. Meng, Extremal problems for eigenvalues of measure differential equations, Proc. Amer. Math. Soc., 143 (2015), 1991-2002.  doi: 10.1090/S0002-9939-2015-12304-0. [7] G. Meng, P. Yan and M. Zhang, Minimization of eigenvalues of one-dimensional p-Laplacian with integrable potentials, J. Optim. Theory Appl., 156 (2013), 294-319.  doi: 10.1007/s10957-012-0125-3. [8] G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Eqautions, 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001. [9] A. B. Mingarelli, Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions Lecture Notes Math., Vol. 989 Springer-Verlag, New York, 1983. [10] P. Savoye, Equimeasurable rearrangements of functions and fourth order boundary value problems, Rocky Mountain J. Math., 26 (1996), 281-293.  doi: 10.1216/rmjm/1181072116. [11] Š. Schwabik, Generalized Ordinary Differential Equations World Scientific, Singapore, 1992. [12] M. Tvrdý, Linear distributional differential equations of the second order, Math. Bohem., 119 (1994), 415-436. [13] M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104. [14] Q. Wei, G. Meng and M. Zhang, Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L1 balls, J. Differential Equations, 247 (2009), 364-400.  doi: 10.1016/j.jde.2009.04.008. [15] P. Yan and M. Zhang, Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. Amer. Math. Soc., 363 (2011), 2003-2028.  doi: 10.1090/S0002-9947-2010-05051-2. [16] M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L1 balls, J. Differential Equations, 246 (2009), 4188-4220.  doi: 10.1016/j.jde.2009.03.016. [17] M. Zhang, Minimization of the zeroth Neumann eigenvalues with integrable potentials, Ann. Inst. H.Poincaré Anal. Non Linéaire, 29 (2012), 501-523.  doi: 10.1016/j.anihpc.2012.01.007.

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##### References:
 [1] M. Carter and B. van Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction Springer-Verlag, New York, 2000. [2] R. Courant and D. Hilbert, Methods of Mathematical Physics Wiley, New York, 1953. [3] Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ., 16 (2009), 299-313. [4] X. Jiang, K. Liu, G. Meng and Z. She, Continuity of the eigenvalues for a vibrating beam, Appl. Math. Lett., 67 (2017), 60-66.  doi: 10.1016/j.aml.2016.12.006. [5] R. E. Megginson, An Introduction to Banach Space Theory Graduate Texts in Mathematics, 183 Springer-Verlag, New York, 1998. [6] G. Meng, Extremal problems for eigenvalues of measure differential equations, Proc. Amer. Math. Soc., 143 (2015), 1991-2002.  doi: 10.1090/S0002-9939-2015-12304-0. [7] G. Meng, P. Yan and M. Zhang, Minimization of eigenvalues of one-dimensional p-Laplacian with integrable potentials, J. Optim. Theory Appl., 156 (2013), 294-319.  doi: 10.1007/s10957-012-0125-3. [8] G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Eqautions, 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001. [9] A. B. Mingarelli, Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions Lecture Notes Math., Vol. 989 Springer-Verlag, New York, 1983. [10] P. Savoye, Equimeasurable rearrangements of functions and fourth order boundary value problems, Rocky Mountain J. Math., 26 (1996), 281-293.  doi: 10.1216/rmjm/1181072116. [11] Š. Schwabik, Generalized Ordinary Differential Equations World Scientific, Singapore, 1992. [12] M. Tvrdý, Linear distributional differential equations of the second order, Math. Bohem., 119 (1994), 415-436. [13] M. Tvrdý, Differential and integral equations in the space of regulated functions, Mem. Differential Equations Math. Phys., 25 (2002), 1-104. [14] Q. Wei, G. Meng and M. Zhang, Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L1 balls, J. Differential Equations, 247 (2009), 364-400.  doi: 10.1016/j.jde.2009.04.008. [15] P. Yan and M. Zhang, Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. Amer. Math. Soc., 363 (2011), 2003-2028.  doi: 10.1090/S0002-9947-2010-05051-2. [16] M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L1 balls, J. Differential Equations, 246 (2009), 4188-4220.  doi: 10.1016/j.jde.2009.03.016. [17] M. Zhang, Minimization of the zeroth Neumann eigenvalues with integrable potentials, Ann. Inst. H.Poincaré Anal. Non Linéaire, 29 (2012), 501-523.  doi: 10.1016/j.anihpc.2012.01.007.
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