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

Bradley, J. S.; Everitt, W. N.

doi:10.1090/S0002-9947-1973-0330606-8

Inequalities associated with regular and singular problems in the calculus of variations
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by J. S. Bradley and W. N. Everitt

Trans. Amer. Math. Soc. 182 (1973), 303-321

DOI: https://doi.org/10.1090/S0002-9947-1973-0330606-8

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Abstract:

An inequality of the form $\smallint _a^b[p|f’{|^2} + q|f{|^2}] \geq {\mu _0}\smallint _a^b|f{|^2}\;(f \in D)$ 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

L. È. Èl′sgol′c, Calculus of variations, Pergamon Press, Ltd., London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1962. MR 0133032
Gilbert A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill., 1946. MR 0017881
John S. Bradley, Adjoint quasi-differential operators of Euler type, Pacific J. Math. 16 (1966), 213–237. MR 200518, DOI 10.2140/pjm.1966.16.213

On the spectrum of ordinary second-order differential operators

68

Methoden der Mathematischen Physik

16

W. N. Everitt, On the limit-point classification of second-order differential operators, J. London Math. Soc. 41 (1966), 531–534. MR 200519, DOI 10.1112/jlms/s1-41.1.531
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 387709
W. N. Everitt, On the spectrum of a second order linear differential equation with a $p$-integrable coefficient, Applicable Anal. 2 (1972), 143–160. MR 397072, DOI 10.1080/00036817208839034
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 326047, DOI 10.1007/BF01428264
I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Daniel Davey & Co., Inc., New York, 1966. Translated from the Russian by the IPST staff; Israel Program for Scientific Translations, Jerusalem, 1965. MR 0190800

Inequalities

Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473

Zur Variationrechung

2

Linear differential operators

16

41

C. R. Putnam, An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (1948), 780–803. MR 30133, DOI 10.2307/2372212
William T. Reid, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0273082
E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
Robert Weinstock, Calculus of variations with applications to physics and engineering, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0052702