Abstract
In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L p topologies, but also with respect to the weak topologies of the L p spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Levitan, B. M., Sargsjan, L. S.: Sturm-Liouville and Dirac Operators, Math. & Appl. (Soviet Ser.), Vol. 59, Kluwer Academic Publishers, Dordrecht, 1991
Pöschel, J., Trubowitz, E.: The Inverse Spectral Theory, Academic Press, New York, 1987
Ortega, R.: The first interval of stability of a periodic equation of Duffing type. Proc. Amer. Math. Soc., 115, 1061–1067 (1992)
Zhang, M.: Continuity in weak topology: higher order linear systems of ODE. Sci. China Ser. A, 51, 1036–1058 (2008)
Gordon, C., Webb, D. L., Wolpert, S.: One cannot hear the shape of a drum. Bull. Amer. Math. Soc. (N. S.), 27, 134–138 (1992)
Chen, M. F.: Spectral gap and logarithmic Sobolev constant for continuous spin systems. Acta Mathematica Sinica, English Series, 24, 705–736 (2008)
Halas, Z., Tvrdý, M.: Approximated solutions of generalized linear differential equations, Preprint, 2008, http://www.math.cas.cz/tvrdy/publ-iso1.html#preprints
Tvrdý, M.: Differential and integral equations in the space of regulated functions. Mem. Differential Equations Math. Phys., 25, 1–104 (2002)
Zhang, M.: Extremal values of smallest eigenvalues of Hill’s operators with potentials in L 1 balls. J. Differential Equations, 246, 4188–4220 (2009)
Wei, Q., Meng, G., Zhang, M.: Extremal values of eigenvalues of Sturm-Liouville operators with potentials in L 1 balls. J. Differential Equations, 247, 364–400 (2009)
Dunford, N., Schwartz, J. T.: Linear Operators, Part I, Interscience, New York, 1958
Gan, S., Zhang, M.: Resonance pockets of Hill’s equations with two-step potentials. SIAM J. Math. Anal., 32, 651–664 (2000)
Zhang, M.: The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. London Math. Soc. (2), 64, 125–143 (2001)
Hale, J. K.: Ordinary Differential Equations, 2nd ed., Wiley, New York, 1969
Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge, 1995
Karaa, S.: Sharp estimates for the eigenvalues of some differential equations. SIAM J. Math. Anal., 29, 1279–1300 (1998)
Krein, M. G.: On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, AMS Translations, Ser. 2, Vol. 1, 1955, 163–187
Zhang, M.: Sobolev inequalities and ellipticity of planar linear Hamiltonian systems. Adv. Nonlinear Stud., 8, 633–654 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The second author is supported by National Basic Research Program of China (Grant No. 2006CB805903) and National Natural Science Foundation of China (Grant Nos. 10325102 and 10531010)
Rights and permissions
About this article
Cite this article
Meng, G., Zhang, M.R. Continuity in weak topology: First order linear systems of ODE. Acta. Math. Sin.-English Ser. 26, 1287–1298 (2010). https://doi.org/10.1007/s10114-010-8103-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-010-8103-x