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Journal of Convex Analysis 24 (2017), No. 1, 123--133
Copyright Heldermann Verlag 2017

On Legendre and Weierstrass Conditions in One-Dimensional Variational Problems

Mikhail A. Sychev
Sobolev Institute for Mathematics, Koptuyg Avenue 4, Novosibirsk 630090, Russia

N. N. Sycheva
Dept. of Mathematics, Novosibirsk State University, Novosibirsk 630090, Russia

We show that classical conditions of Legendre and Weierstrass characterize lower semicontinuity of the correspondent integral functional in appropriate classes of functions. A strengthened version of these conditions characterize the property of the functional of convergence in energy.

Keywords: Integral functionals, Legendre condition, Weierstrass condition, convexity at a point, strict convexity at a point, lower semicontinuity, convergence in energy, Young measures.

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