Math 6410 • ODE

\begin{enumerate}
\item Let \(G\subset\R^N\) be an open set. Define what is meant by a distribution on
\(G\). Let \(f\in L^2(G)\) be a real or complex valued function. Show that \(f\) may be
identified with a distribution \(T_f\).

\item Let \(T\) be a distribution on \(G\) and let
\(\alpha=(\alpha_1,\cdots,\alpha_N)\) be a multiindex. Define the concept of the
\(\alpha\)-th distributional derivative of \(T\).

\item Let
\[
  r:\R^N\to\R
\]
be defined by
\[
\begin{aligned}
  r(x)&=1,\quad \lvert x\rvert<1\\
  r(x)&=0,\quad \lvert x\rvert\ge1,
\end{aligned}
\]
where \(\lvert\cdot\rvert\) is the Euclidean norm in \(\R^N\). Show \(r\) may be thought
of as a distribution on \(\R^N\) and compute its distributional gradient \(\nabla r\).
\end{enumerate}