Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Inequalities associated with regular and singular problems in the calculus of variations

Authors: J. S. Bradley and W. N. Everitt
Journal: Trans. Amer. Math. Soc. 182 (1973), 303-321
MSC: Primary 34B25; Secondary 49B10
MathSciNet review: 0330606
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An inequality of the form $ \smallint _a^b[p\vert f'{\vert^2} + q\vert f{\vert^2}] \geq {\mu _0}\smallint _a^b\vert f{\vert^2}\;(f \in D)$ is established, where p and q are real-valued coefficient functions and f is a complex-valued function in a set D so chosen that both sides of the inequality are finite. The interval of integration is of the form $ - \infty < a < b \leq \infty $. The inequality is first established for functions in the domain of an operator in the Hilbert function space $ {L^2}(a,b)$ that is associated with the differential equation $ - (py')' + qy = \lambda y$, and the number $ {\mu _0}$ in the inequality is the smallest number in the spectrum of this operator. An approximation theorem is given that allows the inequality to be established for the larger set of functions D. An extension of some classical results from the calculus of variations and some spectral theory is then used to give necessary and sufficient conditions for equality and to show that the constant $ {\mu _0}$ is best possible. Certain consequences of these conclusions are also discussed.

References [Enhancements On Off] (What's this?)

  • [1] L. È. Èl′sgol′c, Calculus of variations, Pergamon Press Ltd., London-Paris-Frankfurt, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR 0133032
  • [2] Gilbert A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. MR 0017881
  • [3] John S. Bradley, Adjoint quasi-differential operators of Euler type, Pacific J. Math. 16 (1966), 213–237. MR 0200518
  • [4] J. Chaudhuri and W. N. Evcritt, On the spectrum of ordinary second-order differential operators, Proc. Roy. Soc. Edinburgh A 68 (1968), 95-119.
  • [5] R. Courant and D. Hilbert, Methoden der Mathematischen Physik. Vol. I, Springer, Berlin, 1931; English transl., Interscience, New York, 1953. MR 16, 426.
  • [6] W. N. Everitt, On the limit-point classification of second-order differential operators, J. London Math. Soc. 41 (1966), 531–534. MR 0200519,
  • [7] W. N. Everitt, On an extension to an integro-differential inequality of Hardy, Littlewood and Polya, Proc. Roy. Soc. Edinburgh Sect. A 69 (1972), no. 4, 295–333. MR 0387709
  • [8] W. N. Everitt, On the spectrum of a second order linear differential equation with a 𝑝-integrable coefficient, Applicable Anal. 2 (1972), 143–160. Collection of articles dedicated to Wolfgang Haack on the occasion of his 70th birthday. MR 0397072,
  • [9] W. N. Everitt, M. Giertz, and J. Weidmann, Some remarks on a separation and limit-point criterion of second-order, ordinary differential expressions, Math. Ann. 200 (1973), 335–346. MR 0326047,
  • [10] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Translated from the Russian by the IPST staff, Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey & Co., Inc., New York, 1966. MR 0190800
  • [11] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
  • [12] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [13] L. Lichtenstein, Zur Variationrechung, Kgl. Ges. Wiss. Nach. Math.-Phys. Kl. 2 (1919), 161-192.
  • [14] M. A. Naĭmark, Linear differential operators. Part II, GITTL, Moscow, 1954; English transl., Ungar, New York, 1968. MR 16, 702; MR 41 #7485.
  • [15] C. R. Putnam, An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (1948), 780–803. MR 0030133,
  • [16] William T. Reid, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0273082
  • [17] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151
  • [18] Robert Weinstock, Calculus of variations with applications to physics and engineering, McGraw-Hill Book Company Inc., New York-Toronto-London, 1952. MR 0052702

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B25, 49B10

Retrieve articles in all journals with MSC: 34B25, 49B10

Additional Information

Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society