Math 6410 • ODE

\begin{enumerate}
\item[(a)] Show that if the real symmetric matrix \(A\) has real eigenvalues
\(\lambda_i\) and orthogonal eigenvectors \(\mathbf{x}_i\), then for any vector
\(\mathbf{y}=\sum_i c_i\mathbf{x}_i\), the smallest eigenvalue satisfies
\[
  \lambda_0\le\frac{\mathbf{y}^TA\mathbf{y}}{\mathbf{y}^T\mathbf{y}}.
\]

\item[(b)] Show that the eigenfunctions \(\phi\) and eigenvalues \(-\lambda\) of the problem
\[
  \nabla^2\phi+\lambda\phi=0\quad\text{in a region }D,
\]
with
\[
  \frac{\partial\phi}{\partial n}+\alpha\phi=0\quad\text{on }\partial D,
\]
where \(\partial/\partial n\) is the outward normal derivative, satisfy
\[
  \lambda\int_D \phi^2\dd x
  =\int_D \lvert\nabla\phi\rvert^2\dd x+\alpha\int_{\partial D}\phi^2\dd s.
\]
Assuming that the eigenfunctions \(\phi\) form a complete orthonormal set, derive an
upper bound for the smallest eigenvalue in terms of an appropriate energy integral
(Rayleigh quotient).
\end{enumerate}