Math 6410 • ODE

\textbf{Problem.}
State the Lax-Milgram theorem and give a brief sketch of a proof. Use it to establish the
existence and uniqueness of solutions (in the sense of distributions) of the boundary
value problem
\[
\begin{aligned}
  -\Delta u &= h,\quad \text{in }\Omega,\\
  u &= 0,\quad \text{on }\partial\Omega,
\end{aligned}
\]
where \(h\in L^2(\Omega)\). Also, in your discussion, tell, in what sense the boundary data
are assumed.