Weak local minimizer of a variational integral

For a fixed-endpoint integral on $W^{1,\infty }(a,b)$, an admissible function $u$ is a weak local minimizer if $I(v)\ge I(u)$ whenever the Sobolev norm $\Vert v-u{\Vert }_{W^{1,\infty }}$ is sufficiently small. Both function and derivative must be close. “Weak” here labels the variational neighborhood convention, not the weak topology of a Banach space.

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 $g^{\prime }$ equal to the same nonzero constant at both slopes, and $g^{\prime \prime }>0$ on small neighborhoods of both. For sufficiently small Lipschitz perturbations, local strict convexity bounds the energy difference below by $c\int {\varphi }^{\prime 2}$, since the common first-variation coefficient multiplies $\int {\varphi }^{\prime }=0$. 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.