= Periodic slopes give weak minima without strong minima
{title2=$g(v)=-\cos(\pi v/2),\quad v_1=\delta,\ v_2=\delta-4$}
For $0<\delta<1$, choose the zero-trace piecewise affine curve with slope $\delta$ up to $1-\delta/4$ and slope $\delta-4$ thereafter. Periodicity makes $g'$ equal to the same nonzero constant at both slopes, and $g''>0$ on small neighborhoods of both. For sufficiently small Lipschitz perturbations, local <strict convexity> bounds the energy difference below by $c\int\phi'^2$, since the common first-variation coefficient multiplies $\int\phi'=0$. This proves a strict weak minimum. The global supporting-tangent condition fails because its nonzero slope cannot stay below bounded $g$, so it is not a strong minimum. By contrast, tents of slopes $4n,-4n$ attain $g=-1$ everywhere and provide infinitely many distinct global minimizers.
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