Math 6410 • ODE

\textbf{Variational Principles.}
Let \(\Omega\subset\R^n\) be a bounded domain with smooth boundary \(\partial\Omega\) and
unit outward normal \(\eta\). Consider the problem
\[
\begin{aligned}
  \Delta u-\epsilon u &= f,\quad \text{in }\Omega,\\
  \frac{\partial u}{\partial\eta} &= 0,\quad \text{on }\partial\Omega,
\end{aligned}
\]
where \(f\in L^2(\Omega)\) satisfies \(\int_\Omega f=0\).
\begin{enumerate}
\item[(a)] Prove that for each real constant \(\epsilon>0\), there exists a unique weak
solution \(u_\epsilon\in H^1(\Omega)\).
\item[(b)] In the case \(\epsilon=0\), find an additional condition on \(u\) which guarantees
a unique weak solution \(u_0\), such that the solutions \(u_\epsilon\) from part (a)
converge to \(u_0\) as \(\epsilon\to0^+\). Prove existence and uniqueness of \(u_0\) (with the
extra condition), and prove \(u_\epsilon\to u_0\) in \(H^1(\Omega)\) as \(\epsilon\to0^+\).
\end{enumerate}