= Weierstrass sufficient condition
{c}
Under the usual $C^2$ integrand regularity, an admissible $C^1$ curve satisfying the weak <Euler-Lagrange equation>, a coercively positive <second variation>, and the <strengthened Weierstrass condition> is a <strong local minimizer of a variational integral>. Strict excess away from the reference slope gives strict minimality. Coercive positivity means $\delta^2I(u)[\eta,\eta]\geq c\|\eta\|_{W^{1,2}}^2$ for every zero-trace variation and some $c>0$.
The <second variation> provides a neighboring field of extremals with slope $p(x,y)$. The associated endpoint-invariant Hilbert integral yields $I(v)-I(u)=\int E(x,v,p(x,v),v')\,dx$ for uniformly close admissible $v$. The tube condition makes this integral nonnegative even when $v'$ is large, explaining the passage from weak to strong minimality.
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