= Coercive positivity of a second variation
{title2=$q(\varphi)\geq c\int_a^b|\varphi'|^2$}
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: $q(\varphi)=\int\varphi^2$ is strictly positive, but high-frequency variations make its ratio to $\int\varphi'^2$ tend to zero.
For $q=\int(p\varphi'^2+2r\varphi\varphi'+s\varphi^2)$ with <continuous> coefficients, coercive positivity implies $p\geq c$. 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.
Back to article page
