Quadratic functionals of second order

Author:
Walter Leighton

Journal:
Trans. Amer. Math. Soc. **151** (1970), 309-322

MSC:
Primary 49.00; Secondary 34.00

MathSciNet review:
0264485

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Abstract: In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions with fixed endpoint conditions on and its derivative at and at . Necessary conditions and sufficient conditions are obtained. These lead, in turn, to natural extensions of the Sturm comparison theorem to fourth-order selfadjoint equations. These extensions include and are more general than previously stated comparison theorems. Finally, it is found that the present variational theory provides an orderly approach to second-order Wirtinger-like inequalities, and the results include as special cases many published results of this type.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1970-0264485-1

Keywords:
Quadratic functional of second order,
selfadjoint fourth-order differential equation,
second variation,
minimum of a functional,
Euler equation,
conjugate point,
admissible variation,
comparison theorem,
Wirtinger inequality

Article copyright:
© Copyright 1970
American Mathematical Society