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This paper deals with necessary and sufficient conditions for weak and strongminimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\inC^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler thanthe known ones, and then apply them to several particular problems, includingstability problems in the elasticity theory. In particular, we solve some openproblems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices andbuckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] byfinding optimal conditions for the stability of a naturally straight Kirchhoffrod under various types of endpoint constraints.

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity