Coercive positivity of a second variation ($q(\phi )\ge c{\int }_a^b|{\phi }^{\prime }{|}^2$)

For fixed-endpoint variations, a second variation is coercively positive if the displayed estimate holds for one constant $c>0$ and every admissible variation. The endpoint Poincaré inequality makes this equivalent to a positive lower bound in the $W^{1,2}$ norm. It is stronger than strict positivity on each nonzero variation: $q(\phi )=\int {\phi }^2$ is strictly positive, but high-frequency variations make its ratio to $\int {\phi }^{\prime 2}$ tend to zero.

For $q=\int (p{\phi }^{\prime 2}+2r\phi {\phi }^{\prime }+s{\phi }^2)$ with continuous coefficients, coercive positivity implies $p\ge c$. Indeed, if $p-c<0$ on an interval, multiply a smooth function supported there by $\sin (nx)$. Its negative highest-derivative contribution has order $n^2$, while the mixed term has order at most $n$ and the remaining term is bounded. This contradicts the displayed inequality for large $n$. In particular coercive positivity implies the Strengthened Legendre condition, whereas mere strict positivity does not.