Second variation and the Legendre condition

Use $\delta^2J[u](h,h)$ for the second derivative of $J[u+\varepsilon h]$ at zero, so the Taylor expansion contains one half times this quantity. Assume a twice continuously differentiable integrand and classical candidate and directions on a compact interval, with derivatives bounded in a neighborhood so differentiation under the integral is legitimate. The second derivative consists of a displacement term, a mixed term, and a derivative term. Its associated bilinear form follows by using two different directions or by polarization.

At a two-sided local minimum each scalar restriction has nonnegative second derivative once its first derivative vanishes. Thus the quadratic form must be nonnegative for every admissible direction. Finding a single negative value refutes local minimality: arbitrarily small multiples of that fixed direction lower the objective. A zero value is inconclusive in general because higher-order terms can raise, lower, or leave the objective unchanged. Quadratic functionals are especially transparent because their expansion has no higher-order remainder.

For a smooth scalar one-dimensional fixed-endpoint problem, a classical weak local minimum in the continuously differentiable topology must satisfy $F_{vv}\ge0$ along its graph. This is the Legendre necessary condition. The idea is to localize a perturbation in a region where that coefficient might be negative, making the derivative contribution dominate the lower-order terms. Standard proofs require sufficient smoothness to control these localized estimates; the statement is about the candidate graph and all admissible localized directions, not arbitrary values of the integrand's arguments.

Strictly positive derivative curvature does not by itself make the entire second variation positive. A sufficiently negative displacement coefficient can outweigh it on a long interval. Nor does nonnegative derivative curvature ensure existence of a minimizing curve. In vector-valued or higher-dimensional problems, the correct necessary condition concerns rank-one directions in the gradient Hessian, called the Legendre–Hadamard condition, rather than naive scalar convexity in every matrix direction. The present lesson proves its counterexample in the scalar setting and leaves that generalization as a separate topic.

In a Banach-space formulation, assume the functional is twice Fréchet differentiable near a stationary point and has expansion remainder little-o of the squared norm. If its second derivative is bounded below by a positive constant times the squared norm for every admissible perturbation, the remainder is eventually dominated by that positive term. This proves a strict local minimum in that norm. Mere strict positivity separately for each nonzero direction does not supply such a uniform constant, so it cannot be substituted into this proof.

For a positive quadratic energy on $H_0^1$, Poincaré's inequality can convert a derivative bound into an estimate for the full Sobolev norm. For a nonlinear integral energy, establishing a Fréchet remainder in $H^1$ may require additional growth conditions; classical smoothness on bounded position-slope sets does not control arbitrarily large slopes allowed by small $H^1$ norm. It is legitimate to prove a continuously differentiable local result under classical hypotheses instead, provided that topology and conclusion are stated honestly.