Math 6410 • ODE

Let \(A(x)\) be an \(n\times n\) matrix-valued function defined on the smooth bounded
connected domain \(U\subset\R^n\). Assume \(A(x)=(a_{i,j}(x))\) is uniformly symmetric
positive definite, and \(a_{i,j}\in C^\infty(\overline{U})\) for all \(1\le i,j\le n\).
Consider the problem
\[
\begin{aligned}
  -\operatorname{div}(ADu) &= \lambda u, && \text{in }U,\\
  u &= 0, && \text{on }\partial U.
\end{aligned}
\]
\begin{enumerate}
\item[(a)] Characterize the principal (smallest) eigenvalue \(\lambda_1\) in terms of
the Rayleigh quotient for this problem.
\item[(b)] Prove that \(\lambda_1>0\), and that the associated eigenfunction \(u_1\) is a
minimizer of the Rayleigh quotient.
\end{enumerate}