Advanced Search
Article Contents
Article Contents

Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations

  • * Corresponding author: Bing Xie

    * Corresponding author: Bing Xie

The first author is supported by the NSF of China grants 11771253, 11971262 and 61977043. The second author is supported by NSF of the Shandong Province grants ZR2019MA038, ZR2019MA050 and ZR2017MA049

Abstract Full Text(HTML) Related Papers Cited by
  • The present paper is concerned with the extremal problem of the $ L^1 $-norm of the weights for non-left-definite eigenvalue problems of vibrating string equations with separated boundary conditions. Applying the critical equations of the weights, the infimum is obtained in terms of the given eigenvalue and the parameter in boundary conditions.

    Mathematics Subject Classification: Primary:34B24;Secondary:34L15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] G. Borg, On a Liapunoff criterion of stability, Amer. J. Math., 71 (1949), 67-70.  doi: 10.2307/2372093.
    [2] J. B. Conway, A Course in Functional Analysis, Springer, New York, 1990.
    [3] N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ, Interscience Publishers, New York, 1988.
    [4] Yu. V. Egorov and V. A. Kondrat'ev, Estimates for the First Eigenvalue in some Sturm-Liouville Problems, Russian Math. Surveys, 51 (1996), 439-508. doi: 10.1070/RM1996v051n03ABEH002911.
    [5] H. Guo and J. Qi, Extremal norm of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions, Electron. J. Differential Equations, 99 (2017), 1-11. 
    [6] H. Guo and J. Qi, Sturm-Liouville problems involving distribution weights and an application to optimal problems, J. Optim. Theory Appl., 184 (2019), 842-857.  doi: 10.1007/s10957-019-01584-x.
    [7] D. Hinton and M. Mccarthy, Bounds and optimization of the minimum eigenvalue for a vibrating system, Electron. J. Qual. Theory Differ. Equ., 48 (2013), 1-22.  doi: 10.14232/ejqtde.2013.1.48.
    [8] Y. S. Ilyasov and N. F. Valeeva, On nonlinear boundary value problem corresponding to N-dimensional inverse spectral problem, J. Differential Equations, 266 (2019), 4533-4543.  doi: 10.1016/j.jde.2018.10.003.
    [9] Y. S. Ilyasov and N. F. Valeeva, On an inverse spectral problem and a generalized Sturm's nodal theorem for nonlinear boundary value problems, Ufa Math. J., 10 (2018), 122-128.  doi: 10.13108/2018-10-4-122.
    [10] S. Karaa, Sharp estimates for the eigenvalues of some differential equations, SIAM J. Math. Anal., 29 (1998), 1279-1300.  doi: 10.1137/S0036141096307849.
    [11] 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.  doi: 10.1090/trans2/001/08.
    [12] R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, Vol. 183 doi: 10.1007/978-1-4612-0603-3.
    [13] J. P. Pinasco, Lyapunov-type Inequalities with Application to Eigenvalue Problems, Spinger Briefs in Mathematics, 2013. doi: 10.1007/978-1-4614-8523-0.
    [14] J. Pöschel and  E. TrubowitzThe Inverse Spectral Theory, Academic Press, New York, 1987. 
    [15] J. Qi and S. Chen, Extremal norms of the potentials recovered from inverse Dirichlet problems, Inverse Problems, 32 (2016), 1-13.  doi: 10.1088/0266-5611/32/3/035007.
    [16] Q. WeiG. Meng and M. Zhang, Extremal values of eigenvalues of Sturm-Liouville operators with potentials in $L^1$ balls, J. Differential Equations, 247 (2009), 364-400.  doi: 10.1016/j.jde.2009.04.008.
    [17] Z. Wen and M. Zhang, On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures, Discrete Contin. Dyn. Syst. B, 25 (2020), 3257-3274.  doi: 10.3934/dcdsb.2020061.
    [18] M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in $L^1$ balls, J. Differential Equations, 246 (2009), 4188-4220.  doi: 10.1016/j.jde.2009.03.016.
    [19] M. ZhangZ. WenG. MengJ. Qi and B. Xie, On the number and complete continuity of weighted eigenvalues of measure differential equations, Differential Integral Equations, 31 (2018), 761-784. 
  • 加载中

Article Metrics

HTML views(1743) PDF downloads(287) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint