I have been studying variational principles and I have been reading this set of notes. In section 7.1, we study the Sturm-Liouville problem, as described below.
Let $p(x)$, $\sigma(x)$, $w(x)$ be real functions of $x$ defined for $\alpha \leq x \leq \beta$ with $p$ and $w$ positive for $\alpha \lt x \lt \beta$ and consider the following real functionals of the real function $y(x)$: $$F[y]=\int_\alpha^\beta p(x)(y')^2+\sigma(x)y^2dx,$$ $$G[y]=\int_\alpha^\beta w(x)y^2dx.$$ The problem is to find the function $y$ that minimised $F[y]$ subject to $G[y]=1$, given that $y(\alpha)$ and $y(\beta)$ are fixed.
The E-L equations become $$Ly=\lambda wy,$$ where $L$ is the differential operator $$L=-\frac{d}{dx}\left(p(x)\frac{d}{dx}\right)+\sigma(x).$$ Now suppose $\sigma(x)$ is positive so that $F \geq 0$, and therefore has a positive minimum. We have $$\lambda G=\int_\alpha^\beta yLydx=F-[pyy']_{\alpha}^{\beta}$$ and the lecturer now claims that the boundary term is zero so that $\lambda = F/G \geq0$. Since $F[Ay]=A^2F[y]$ and $G[Ay]=A^2G[y]$ for any constant $A$ and function $y$ such that $F[y]$ and $G[y]$ exist, the original problem is equivalent to minimising $F/G$ so the lowest eigenvalue of $L$ will be the minimum of $F/G$.
I have two questions:
1) Why does the boundary term disappear as claimed? Do we not also need to specify that $y(\alpha)=y(\beta)=0$ or impose some other appropriate constraint?
2) Why is it necessary to specify that $p$ and $w$ are positive and then later to also specify that $\sigma$ is positive? Where do we use these assumptions and why is the specification that $\sigma$ is positive not needed until later?