Euler-Lagrange is necessary but not sufficient without convexity

For a Gateaux differentiable functional on an open set, the first variation vanishes at an interior local minimiser; on an affine admissible class $u+V$ it vanishes in the directions $v\in V$ (The first variation vanishes at an interior minimiser). For integral functionals satisfying its differentiation and fixed-trace hypotheses, The weak Euler-Lagrange equation for integral functionals with fixed trace gives the weak Euler-Lagrange equation for zero-boundary variations. Conversely, for a convex functional Gateaux differentiable on an open neighbourhood of a convex admissible set $K$, the condition $\delta I(u;v-u)\ge 0$ for every $v\in K$ suffices for a global minimum (Stationarity is sufficient for a global minimum of a convex differentiable functional). Without convexity the three notions must be kept apart: a stationary point solves the Euler-Lagrange equation, a local minimiser minimises among nearby admissible competitors, and a global minimiser minimises on the whole admissible set. In general none of the implications "stationary $\Rightarrow$ local minimiser", "local minimiser $\Rightarrow$ global minimiser" or "global minimiser $\Rightarrow$ unique" holds, and the Euler-Lagrange equation alone therefore cannot be used as an existence criterion. Convexity upgrades the variational inequality to global minimality, whereas coercivity and weak lower semicontinuity enter the separate existence argument; a concave quadratic functional is the standard illustration, and the companion page records explicit counterexamples. For the other failed implications already mentioned, $F(t)=t^2-3t^3+t^4$ has a local minimum at $0$ (the coefficient $1-3t+t^2$ is positive near $0$) but $F(1)=-1<F(0)=0$, while $(t^2-1{)}^2$ has the two global minimisers $\pm 1$.

For inequality constraints, two-sided variations need not be admissible. Local minimality gives only a nonnegative one-sided derivative along an admissible segment, since $[I(u+tv)-I(u)]\mathrm{/}t\ge 0$ for sufficiently small feasible $t>0$; it need not give stationarity in arbitrary directions.