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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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