Math 6410 • ODE

\textbf{Problem.}
Let \(\Omega\) be an open subset of \(\mathbb{R}^N\).
\begin{enumerate}
\item[(1)] Define what is meant by a distribution on \(\Omega\).
\item[(2)] Show that all functions in \(L^1_{\mathrm{loc}}(\Omega)\), are, by an
appropriate identification, distributions.
\item[(3)] Give an example of a distribution which is not of this type.
\item[(4)] Give the derivative formula for differentiating distributions.
\item[(5)] Give a definition of the \(L^2\) Sobolev spaces \(H^m(\Omega)\) and
\(H^m_0(\Omega)\). (Other notations used are the \(L^2\) Sobolev spaces \(W^{m,2}(\Omega)\) and
\(W^{m,2}_0(\Omega)\).)
\end{enumerate}