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), 303321
MSC:
Primary 34B25; Secondary 49B10
MathSciNet review:
0330606
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Abstract: An inequality of the form is established, where p and q are realvalued coefficient functions and f is a complexvalued function in a set D so chosen that both sides of the inequality are finite. The interval of integration is of the form . The inequality is first established for functions in the domain of an operator in the Hilbert function space that is associated with the differential equation , and the number 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 is best possible. Certain consequences of these conclusions are also discussed.
 [1]
L.
È. Èl′sgol′c, Calculus of
variations, Pergamon Press Ltd., LondonParisFrankfurt,
AddisonWesley Publishing Co., Inc., Reading, Mass., 1962. MR 0133032
(24 #A2868)
 [2]
Gilbert
A. Bliss, Lectures on the Calculus of Variations, University
of Chicago Press, Chicago, Ill., 1946. MR 0017881
(8,212e)
 [3]
John
S. Bradley, Adjoint quasidifferential operators of Euler
type, Pacific J. Math. 16 (1966), 213–237. MR 0200518
(34 #409)
 [4]
J. Chaudhuri and W. N. Evcritt, On the spectrum of ordinary secondorder differential operators, Proc. Roy. Soc. Edinburgh A 68 (1968), 95119.
 [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 limitpoint classification of secondorder
differential operators, J. London Math. Soc. 41
(1966), 531–534. MR 0200519
(34 #410)
 [7]
W.
N. Everitt, On an extension to an integrodifferential inequality
of Hardy, Littlewood and Polya, Proc. Roy. Soc. Edinburgh Sect. A
69 (1972), no. 4, 295–333. MR 0387709
(52 #8548)
 [8]
W.
N. Everitt, On the spectrum of a second order linear differential
equation with a 𝑝integrable coefficient, Applicable Anal.
2 (1972), 143–160. Collection of articles dedicated
to Wolfgang Haack on the occasion of his 70th birthday. MR 0397072
(53 #932)
 [9]
W.
N. Everitt, M.
Giertz, and J.
Weidmann, Some remarks on a separation and limitpoint criterion of
secondorder, ordinary differential expressions, Math. Ann.
200 (1973), 335–346. MR 0326047
(48 #4393)
 [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,
1966. MR
0190800 (32 #8210)
 [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, SpringerVerlag
New York, Inc., New York, 1966. MR 0203473
(34 #3324)
 [13]
L. Lichtenstein, Zur Variationrechung, Kgl. Ges. Wiss. Nach. Math.Phys. Kl. 2 (1919), 161192.
 [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
(10,718i)
 [16]
William
T. Reid, Ordinary differential equations, John Wiley &
Sons Inc., New York, 1971. MR 0273082
(42 #7963)
 [17]
E.
C. Titchmarsh, Eigenfunction expansions associated with
secondorder differential equations. Part I, Second Edition, Clarendon
Press, Oxford, 1962. MR 0176151
(31 #426)
 [18]
Robert
Weinstock, Calculus of variations with applications to physics and
engineering, McGrawHill Book Company Inc., New YorkTorontoLondon,
1952. MR
0052702 (14,661a)
 [1]
 N. I. Ahiezer, The calculus of variations, GITTL, Moscow, 1955; English transl., Blaisdell, Waltham, Mass., 1962. MR 17, 861; MR 25 #5414. MR 0133032 (24:A2868)
 [2]
 G. A. Bliss, Lectures on the calculus of variations, Univ. of Chicago Press, Chicago, Ill., 1946. MR 8, 212. MR 0017881 (8:212e)
 [3]
 J. S. Bradley, Adjoint quasidifferential operators of Euler type, Pacific J. Math. 16 (1966), 213237. MR 34 #409. MR 0200518 (34:409)
 [4]
 J. Chaudhuri and W. N. Evcritt, On the spectrum of ordinary secondorder differential operators, Proc. Roy. Soc. Edinburgh A 68 (1968), 95119.
 [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 limitpoint classification of second order differential operators, J. London Math. Soc. 41 (1966), 531534. MR 34 #410. MR 0200519 (34:410)
 [7]
 , On an extension to an integrodifferential inequality of Hardy, Littlewood and Pólya, Proc. Roy. Soc. Edinburgh A 69 (1972), 295333. MR 0387709 (52:8548)
 [8]
 , On the spectrum of a second order linear differential equation with a pintegrable coefficient, Appl. Anal. 2 (1972), 143160. MR 0397072 (53:932)
 [9]
 W. N. Everitt, M. Giertz and J. Weidmann, Some remarks on a limitpoint and separation criterion for secondorder, ordinary differential expressions, Math. Ann. 200 (1973), 335346. MR 0326047 (48:4393)
 [10]
 I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Fizmatgiz, Moscow 1963; English transl., Israel Program for Scientific Translations, Jerusalem, 1965; Davey, New York, 1966. MR 32 #2938; #8210. MR 0190800 (32:8210)
 [11]
 G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
 [12]
 T. Kato, Perturbation theory for linear operators, Die Grundlehren der math. Wissenschaften, Band 132, SpringerVerlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
 [13]
 L. Lichtenstein, Zur Variationrechung, Kgl. Ges. Wiss. Nach. Math.Phys. Kl. 2 (1919), 161192.
 [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), 780803. MR 10, 718. MR 0030133 (10:718i)
 [16]
 W. T. Reid, Ordinary differential equations, Wiley, New York, 1971. MR 42 #7963. MR 0273082 (42:7963)
 [17]
 E. C. Titchmarsh, Eigenfunction expansions associated with secondorder differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 31 #426. MR 0176151 (31:426)
 [18]
 R. Weinstock, Calculus of variations, with applications to physics and engineering, McGrawHill, New York, 1952. MR 14, 661. MR 0052702 (14:661a)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303306068
PII:
S 00029947(1973)03306068
Article copyright:
© Copyright 1973 American Mathematical Society
