• Previous Article
    Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces
  • DCDS Home
  • This Issue
  • Next Article
    On the periodic approximation of Lyapunov exponents for semi-invertible cocycles
December  2017, 37(12): 6369-6382. doi: 10.3934/dcds.2017276

The optimal upper bound for the first eigenvalue of the fourth order equation

Schinftyl of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received  February 2017 Published  August 2017

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No. 11201471 and 11671378) and the Fund of UCAS.

In this paper we will give the optimal upper bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1 norm of potentials is known. Combining with the results for the corresponding optimal lower bound problem in [12], we have completely obtained the optimal estimation for the first eigenvalue of the fourth order equation.

Citation: Gang Meng. The optimal upper bound for the first eigenvalue of the fourth order equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6369-6382. doi: 10.3934/dcds.2017276
References:
[1]

B. Andrews and J. Clutterbuck, Prinftyf of the fundamental gap conjecture, J. Amer. Math. Soc., 24 (2011), 899-916.  doi: 10.1090/S0894-0347-2011-00699-1.

[2]

M. van den Berg, On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys., 31 (1983), 623-637.  doi: 10.1007/BF01019501.

[3]

M. Carter and B. van Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1174-7.

[4]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley, New York, 1953.

[5]

C.-Y. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335.  doi: 10.3934/mbe.2008.5.315.

[6]

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

[7]

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. Ser. 2, 1 (1955), 163-187.  doi: 10.1090/trans2/001/08.

[8]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633.  doi: 10.1090/S0002-9947-2010-04931-1.

[9]

T. J. Mahar and B. E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math., 29 (1976), 517-529.  doi: 10.1002/cpa.3160290505.

[10]

R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, 183 Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0603-3.

[11]

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.

[12]

G. Meng and P. Yan, Optimal lower bound for the first eigenvalue of the fourth order equation, J. Differential Eqautions, 261 (2016), 3149-3168.  doi: 10.1016/j.jde.2016.05.018.

[13]

G. MengP. 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.

[14]

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.

[15]

G. A. MonteiroU. M. Hanung and M. Tvrdý, Bounded convergence theorem for abstract Kurzweil-Stieltjes integral, Monatsh. Math., 180 (2015), 1-26.  doi: 10.1007/s00605-015-0774-z.

[16]

Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. doi: 10.1142/1875.

[17]

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.

[18]

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.

[19]

S. T. Yau, Nonlinear Analysis in Geometry, Monographies de L'Enseignement Mathématique, vol. 33, L'Enseignement Mathématique, Geneva, 1986. Série des Conférences de I'Union Mathématique Internationale, 8.

[20]

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.

[21]

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.

show all references

References:
[1]

B. Andrews and J. Clutterbuck, Prinftyf of the fundamental gap conjecture, J. Amer. Math. Soc., 24 (2011), 899-916.  doi: 10.1090/S0894-0347-2011-00699-1.

[2]

M. van den Berg, On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys., 31 (1983), 623-637.  doi: 10.1007/BF01019501.

[3]

M. Carter and B. van Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1174-7.

[4]

R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley, New York, 1953.

[5]

C.-Y. KaoY. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains, Math. Biosci. Eng., 5 (2008), 315-335.  doi: 10.3934/mbe.2008.5.315.

[6]

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

[7]

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. Ser. 2, 1 (1955), 163-187.  doi: 10.1090/trans2/001/08.

[8]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Trans. Amer. Math. Soc., 362 (2010), 5605-5633.  doi: 10.1090/S0002-9947-2010-04931-1.

[9]

T. J. Mahar and B. E. Willner, An extremal eigenvalue problem, Comm. Pure Appl. Math., 29 (1976), 517-529.  doi: 10.1002/cpa.3160290505.

[10]

R. E. Megginson, An Introduction to Banach Space Theory, Graduate Texts in Mathematics, 183 Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0603-3.

[11]

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.

[12]

G. Meng and P. Yan, Optimal lower bound for the first eigenvalue of the fourth order equation, J. Differential Eqautions, 261 (2016), 3149-3168.  doi: 10.1016/j.jde.2016.05.018.

[13]

G. MengP. 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.

[14]

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.

[15]

G. A. MonteiroU. M. Hanung and M. Tvrdý, Bounded convergence theorem for abstract Kurzweil-Stieltjes integral, Monatsh. Math., 180 (2015), 1-26.  doi: 10.1007/s00605-015-0774-z.

[16]

Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. doi: 10.1142/1875.

[17]

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.

[18]

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.

[19]

S. T. Yau, Nonlinear Analysis in Geometry, Monographies de L'Enseignement Mathématique, vol. 33, L'Enseignement Mathématique, Geneva, 1986. Série des Conférences de I'Union Mathématique Internationale, 8.

[20]

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.

[21]

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.

[1]

Monika Laskawy. Optimality conditions of the first eigenvalue of a fourth order Steklov problem. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1843-1859. doi: 10.3934/cpaa.2017089

[2]

Yuyan Yao, Gang Wang. Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors. Mathematical Foundations of Computing, 2022, 5 (1) : 33-44. doi: 10.3934/mfc.2021018

[3]

Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032

[4]

Jiantao Jiang, Jing An, Jianwei Zhou. A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022066

[5]

S. E. Kuznetsov. An upper bound for positive solutions of the equation \Delta u=u^\alpha. Electronic Research Announcements, 2004, 10: 103-112.

[6]

Craig Cowan, Pierpaolo Esposito, Nassif Ghoussoub. Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1033-1050. doi: 10.3934/dcds.2010.28.1033

[7]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial and Management Optimization, 2020, 16 (1) : 309-324. doi: 10.3934/jimo.2018153

[8]

Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080

[9]

Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4551-4574. doi: 10.3934/dcdss.2021113

[10]

Mohammed Mesk, Ali Moussaoui. On an upper bound for the spreading speed. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3897-3912. doi: 10.3934/dcdsb.2021210

[11]

Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

[12]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[13]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure and Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843

[14]

Editorial Office. Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3785-3785. doi: 10.3934/cpaa.2020167

[15]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205

[16]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2589-2618. doi: 10.3934/dcds.2017111

[17]

Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure and Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675

[18]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493

[19]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617

[20]

Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (244)
  • HTML views (72)
  • Cited by (4)

Other articles
by authors

[Back to Top]