doi: 10.3934/dcdsb.2020243

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

School of Mathematics and Statistics, Shandong University, Weihai 264209, China

* Corresponding author: Bing Xie

Received  November 2019 Published  August 2020

Fund Project: 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

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.

Citation: Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020243
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. Karaa, Sharp estimates for the eigenvalues of some differential equations, SIAM J. Math. Anal., 29 (1998), 1279-1300.  doi: 10.1137/S0036141096307849.  Google Scholar

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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.  Google Scholar

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R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, Vol. 183 doi: 10.1007/978-1-4612-0603-3.  Google Scholar

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J. P. Pinasco, Lyapunov-type Inequalities with Application to Eigenvalue Problems, Spinger Briefs in Mathematics, 2013. doi: 10.1007/978-1-4614-8523-0.  Google Scholar

[14] J. Pöschel and E. Trubowitz, The Inverse Spectral Theory, Academic Press, New York, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

G. Borg, On a Liapunoff criterion of stability, Amer. J. Math., 71 (1949), 67-70.  doi: 10.2307/2372093.  Google Scholar

[2]

J. B. Conway, A Course in Functional Analysis, Springer, New York, 1990.  Google Scholar

[3]

N. Dunford and J. T. Schwartz, Linear Operators, Part Ⅰ, Interscience Publishers, New York, 1988.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Karaa, Sharp estimates for the eigenvalues of some differential equations, SIAM J. Math. Anal., 29 (1998), 1279-1300.  doi: 10.1137/S0036141096307849.  Google Scholar

[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.  Google Scholar

[12]

R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, Vol. 183 doi: 10.1007/978-1-4612-0603-3.  Google Scholar

[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.  Google Scholar

[14] J. Pöschel and E. Trubowitz, The Inverse Spectral Theory, Academic Press, New York, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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