Math 6410 • ODE

Let \(D\subset\R^n\) be a bounded, connected domain with smooth boundary. Let \(\phi\)
be an eigenfunction for the Laplacian on \(D\) with Dirichlet Boundary conditions
(\(\phi=0\) on \(\partial D\)) and with eigenvalue \(\lambda\)
\[
  \Delta u+\lambda u=0.
  \tag{2}
\]
\begin{enumerate}
\item[(a)] Show that there is a finite constant \(C\) so that every nonzero continuously
differentiable function \(u\in C^1(D)\) such that \(u=0\) on \(\partial D\) satisfies
\[
  \int_D u^2 \le C\int_D \abs{\nabla u}^2.
\]
\item[(b)] By integrating eigenfunctions of (2), conclude that eigenvalues are
positive \(\lambda>0\).
\item[(c)] Denote the least value of the Rayleigh Quotient
\[
  \lambda_1(D)=
  \inf_{\substack{u\in C_0^1(D)\\ u\not\equiv0}}
  \frac{\int_D \abs{\nabla u}^2}{\int_D u^2}.
\]
which satisfies \(\lambda\ge 1/C\) by (a). 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}