Math 6410 • ODE

\begin{enumerate}
\item Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(m=1,2,\cdots\),
and \(H_0^m(G)\), \(m=1,2,\cdots\). Indicate why these spaces are Hilbert spaces.

\item Let \(G\) be a bounded open set. State and prove Poincar\'e's inequality for
functions \(u\in H_0^1(G)\) and thus verify that \(H_0^1(G)\) is also a Hilbert space
with respect to the inner product
\[
  (u,v)_{H_0^1(G)}=\int_G \nabla u\cdot\overline{\nabla v}\,dx.
\]
\end{enumerate}