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
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

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

[1]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[2]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[3]

Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229

[4]

Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388

[5]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[6]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[7]

Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger Equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020392

[8]

Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227

[9]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101

[10]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[11]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[12]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[13]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[14]

Jaume Llibre, Claudia Valls. Rational limit cycles of abel equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021007

[15]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006

[16]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378

[17]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[18]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[20]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (33)
  • HTML views (184)
  • Cited by (0)

Other articles
by authors

[Back to Top]