Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Mikhail A. Sychev

Sobolev Institute for Mathematics, Koptuyg Avenue 4, Novosibirsk 630090, Russia
and: Novosibirsk State University, Novosibirsk 630090, Russia

masychev@math.nsc.ru

N. N. Sycheva

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

Keywords

  • Integral functionals
  • Legendre condition
  • Weierstrass condition
  • convexity at a point
  • strict convexity at a point
  • lower semicontinuity
  • convergence in energy
  • Young measures

Suggested citation

M. A. Sychev, N. N. Sycheva. “On Legendre and Weierstrass Conditions in One-Dimensional Variational Problems.” Journal of Convex Analysis 24 (2017), No. 1, 123–133. https://doi.org/10.68381/jca24010

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.