= Regularized Jacobi criterion for a degenerate second variation
{c}
{title2=$q_\eta=q+\eta\int|\varphi'|^2$}
For continuous $P\geq0,Q$, a scalar <second variation> $q=\int(P\varphi'^2+Q\varphi^2)$ is strictly positive on nonzero fixed-endpoint <Lipschitz> variations if and only if every regularized form $q_\eta$ for $\eta>0$ has no <conjugate point> in $(a,b]$, and the original weak Jacobi boundary problem has no nonzero <Lipschitz> solution. The <Jacobi positivity criterion> applies to each regularized coefficient $P+\eta>0$. Letting $\eta$ decrease to zero gives $q\geq0$. For a nonnegative <quadratic form>, $q(\varphi)=0$ forces its mixed <bilinear form> with every <test function> to vanish, by considering $q(\varphi+t\psi)$ for both signs of $t$. This is the weak <Jacobi equation>, so the kernel condition gives strict positivity. Necessity follows by the same argument in reverse. It distinguishes strict positivity from a uniform coercive estimate when the <derivative> coefficient degenerates.
Back to article page
