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 PDF

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

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.

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