and let $H^{-1}(a,b):=(H^1_0(a,b))^*$ be its [dual space](/page/Dual%20Space). In the regular Dirichlet Sturm-Liouville setting, define the quadratic form
\begin{align*}
A h(\phi):=\int_a^b \left(P(x)h'(x)\phi'(x)+Q(x)h(x)\phi(x)\right)\,d\mathcal{L}^1(x)
\end{align*}
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for $h,\phi\in H^1_0(a,b)$. Suppose the regular Dirichlet Sturm-Liouville spectral theorem applies to this operator, so that its first Dirichlet eigenvalue is characterized by the Rayleigh quotient and, if the Rayleigh infimum is $0$, then $0$ is a Dirichlet eigenvalue. Then the associated quadratic form extends continuously to $H^1_0(a,b)$ and there exists $\gamma>0$ such that
Finally, suppose the second-order Taylor expansion of $J$ at $y$ is uniform in the following sense: there exist $\rho>0$ and a function $\varepsilon:(0,\rho]\to [0,\infty)$ with $\varepsilon(r)\to 0$ as $r\downarrow 0$ such that, for every $h\in C^1([a,b];\mathbb{R})$ with $h(a)=h(b)=0$, $\|h\|_{C^1([a,b])}\leq \rho$, and $y+h\in \mathcal{A}_{\alpha,\beta}$,