Math 6410 • ODE

Consider the Dirichlet problem
\[
\Delta u=f\quad\text{in }\Omega,
\]
\[
u=0\quad\text{on }\partial\Omega,
\]
where \(\Omega\subset\R^n\) is a bounded open set and \(f\in L_2(\Omega)\).

\begin{enumerate}
\item[(a)] Define the concept of a weak \(H_0^1(\Omega)\) solution of the above
problem.

\item[(b)] Use the Poincar\'e inequality,
\[
\int_\Omega u^2\,dx\le c\int_\Omega\lvert\nabla u\rvert^2\,dx
\quad\text{for all }u\in H_0^1(\Omega),
\]
to prove the existence of a weak \(H_0^1(\Omega)\) solution.
\end{enumerate}