Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2007 / B / 7 / a

A weak local minimizer of a variational integral has some $\rho >0$ for which $I(v)\ge I(\bar{u})$ whenever $v$ is admissible and $\Vert v-\bar{u}{\Vert }_{W^{1,\infty }}<\rho$.

A strong local minimizer of a variational integral has the same property for all admissible $v$ with $\Vert v-\bar{u}{\Vert }_{\infty }<\rho$; its derivative need not be close. Since the supremum norm is bounded by the Sobolev norm, this neighborhood contains a sufficiently small weak neighborhood. Thus every strong local minimizer is a weak local minimizer of a variational integral.

A global minimizer satisfies $I(\bar{u})\le I(v)$ for every admissible $v$.

A point of convexity $s$ of a differentiable scalar function is one where its tangent supports the entire graph: $g(q)\ge g(s)+g^{\prime }(s)(q-s)$ for every $q\in \mathbb{R}$. The Weierstrass necessary condition for strong minimality, under its differentiability assumptions in the slope variable, isfor almost every $x$. In other words the reference slope is a point of convexity of the slope integrand. The later example is smooth, so these derivative-based conditions apply there.