Quadratic functionals of second order

Leighton, Walter

doi:10.1090/S0002-9947-1970-0264485-1

Quadratic functionals of second order
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Trans. Amer. Math. Soc. 151 (1970), 309-322 Request permission

Abstract:

In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions $y(x)$ with fixed endpoint conditions on $y(x)$ and its derivative at $x = a$ and at $x = b$. 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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