Math 6410 • ODE

\begin{enumerate}
\item Let \(G\) be an open subset of \(\R^N\) and let \(f\in L^2(G)\). Show that \(f\)
defines a continuous linear functional on \(H^1(G)\).

\item State the Lax-Milgram theorem for subspaces \(V\subset H^1(G)\).

\item Show that the Lax-Milgram theorem may be applied to the form
\[
  a(u,v)=\sum_{i=1}^{N}\int_G \partial_i u\,\partial_i\overline{v}
  +\int_G u\overline{v}
\]
and \(f\) as above.

Assume that \(G\) is a bounded open set with ``very smooth'' boundary and let \(u\) be
the Lax-Milgram solution just determined. What partial differential equation (in
the sense of distributions) will be solved by \(u\) and what boundary conditions does
\(u\) satisfy (please give an argument to prove the latter)?
\end{enumerate}