Math 6410 • ODE

Let \(\Omega\) be a bounded domain in \(\R^n\). Consider
\[
\begin{aligned}
  \Delta u&=f &&\text{in }\Omega,\\
  u&=0 &&\text{on }\partial\Omega
\end{aligned}
\]
where \(f\in L^2(\Omega)\).
\begin{enumerate}
\item Define the concept of a weak \(H^{1,2}_0(\Omega)\) solution to the above problem.
\item Use Poincar\'e's inequality to prove existence of a weak \(H^{1,2}_0(\Omega)\)
solution.
\item Assume that \(\Omega\) is smooth. Prove that any two solutions in
\(C^2(\Omega)\cap C^1(\overline{\Omega})\) must agree. How would you prove this result for
any two solutions in \(C^2(\Omega)\cap C(\overline{\Omega})\)?
\end{enumerate}