Math 6410 • ODE

Let \(D\in\R^n\) be a bounded, connected domain with smooth boundary. Let \(\phi\) be an
eigen-funcion of the Laplacian on \(D\) with with Dirichlet boundary conditions
(\(\phi=0\) on \(\partial D\)) and with eigenvalue \(\lambda\),
\begin{equation}
  \Delta\phi+\lambda\phi=0.
  \tag{5}
\end{equation}
\begin{enumerate}
\item[(a)] Show that there is a finite constant \(C\) so that for every nonzero
continuously differentiable \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies
\[
  \int_D u^2\le C\int_D \lvert Du\rvert^2.
\]
\item[(b)] By integrating eigenfunctions of (5), conclude that eigenvalues are positive
\(\lambda>0\).
\item[(c)] Denote by
\[
  \lambda_1(D)=
  \inf_{\substack{u\in C_0^1(D),\ u\not\equiv0}}
  \frac{\int_D \lvert Du\rvert^2}{\int_D u^2}
\]
the least value of the Rayleigh quotient (which is at least \(1/C\)). Assuming that
\(\lambda_1(D)\) is an eigenvalue of \(D\) with Dirichlet boundary conditions, show that
if two such domains satisfy \(D_1\subset D_2\), then
\(\lambda_1(D_1)\ge\lambda_1(D_2)\).
\end{enumerate}