Math 6410 • ODE

\begin{enumerate}
\item Let \(\Omega\subset\R^n\) be open and let \(u\in C(\Omega)\). Define what it means to
say that \(u\) is subharmonic in \(\Omega\).
\item Let \(\Omega\) be open, bounded and connected and let
\(u\in C^2(\Omega)\cap C^0(\bar{\Omega})\) be subharmonic. State the weak form of the
maximum principle for \(u\).
\item Consider the Dirichlet problem
\[
\begin{aligned}
  \Delta u &= f && \text{in }\Omega,\\
  u &= 0 && \text{on }\partial\Omega
\end{aligned}
\tag{9.1}
\]
where \(f\in C^0(\Omega)\) and \(\Omega\) is as in (b). Describe in as much detail as you
can the major steps in proving the existence of a classical solution of (9.1) via
Perron's Method.
\end{enumerate}