= Solution
A <global minimizer> has $I(\bar u)\leq I(v)$ for every admissible $v$. Global minimality implies strong local minimality, which implies weak local minimality. The reverse implications are not valid in general.
Where $f_v$ exists, the <strengthened Weierstrass condition> says that for some $\delta>0$, the excess
$$
E(x,y,p,q)=f(x,y,q)-f(x,y,p)-f_v(x,y,p)(q-p)
$$
is nonnegative for every $x\in[a,b]$, every trial slope $q\in\mathbb R$, and all $y,p$ with $|y-\bar u(x)|<\delta$, $|p-\bar u'(x)|<\delta$. This is a neighborhood condition in height and reference slope, not just an inequality on the reference curve.
The classical <Weierstrass sufficient condition>, with the usual smooth integrand regularity in addition to continuity, states that an admissible $C^1$ curve satisfying the <Euler-Lagrange equation>, a coercively positive <second variation>, and this strengthened condition is a strong local minimizer. A strict excess away from the reference slope gives the corresponding strict conclusion. The differentiability assumptions are part of that classical sufficient theorem; continuity alone does not define its first and second variations.
Back to article page
