Math 6410 • ODE

Let \(U\subset\R^n\) be a bounded open set with smooth boundary \(\partial U\), and
consider the problem
\[
\begin{cases}
  -\Delta u+c(x)u=\lambda u+f, & \text{in }U,\\
  u=0, & \text{on }\partial U,
\end{cases}
\]
where \(c(x)\in L^\infty(U)\), and \(f\in C_0^\infty(\partial U)\).
\begin{enumerate}
\item[(a)] In the case \(f=0\), by examining the bilinear form over \(H_0^1(U)\) associated
with the operator \(-\Delta+c-\lambda\), determine a general condition on
\(\lambda\in\R\) which guarantees that \(u=0\) is the only weak solution of the problem.
\item[(b)] Still assuming \(f=0\), suppose that \(\lambda=\lambda_1,\lambda_2\) have been
chosen such that there exist nonzero weak solutions \(u_1,u_2\in H_0^1(U)\) respectively
(that is, \((\lambda_1,u_1)\) and \((\lambda_2,u_2)\) are eigenpairs). Prove that
\(\lambda_j\) must be real, and if \(\lambda_1\ne\lambda_2\), then \(u_1\) and \(u_2\) are
orthogonal.
\item[(c)] Suppose \(\lambda_1\) is an eigenvalue as in part (b), and let
\(E_1\subset H_0^1(U)\) be the set of all associated eigenfunctions. Setting
\(\lambda=\lambda_1\), for general \(f\in C_0^\infty(U)\), find the least restrictive
constraint on \(f\) which guarantees a weak solution to the problem exists. Is the
solution unique?
\end{enumerate}