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.
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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
