Math 6410 • ODE

\textbf{Laplace Equation.}
Let \(\Omega\subset\R^n\) be a domain. Let \(\Omega_1\) be a bounded subdomain with
\(\overline{\Omega}_1\subset\Omega\).
\begin{enumerate}
\item[(a.)] State the Harnack Inequality for harmonic functions.
\item[(b.)] Let \(\{u_n\}\) be a monotone increasing sequence of nonnegative harmonic
functions on \(\Omega\). Suppose that for some point \(\xi\in\Omega_1\), the sequence of real
numbers \(\{u_n(\xi)\}\) is bounded. Prove that \(\{u_n\}\) converges uniformly on
\(\Omega_1\) to a harmonic function. You may use standard theorems about convergence of
sequences of harmonic functions without proof.
\end{enumerate}