Math 6420 • PDE

Consider the problem \(Lu=1\) on \((0,\pi)\), where \(Lu=u''+u\) with boundary conditions
\(u'(0)=u'(\pi)=0\).
\begin{enumerate}
\item[(a)] Write down the weak formulation of the problem.

\item[(b)] Find the \(L^2[0,\pi]\) adjoint of \(L\), and determine whether or not \(L\) is
self-adjoint.

\item[(c)] Find the nullspace of \(L\). Use Fredholm theory to characterize the set of functions
\(f\) such that \(Lu=f\) has a solution. Verify that \(f=1\) satisfies your condition, and find an
infinite set of solutions to the problem \(Lu=1\).
\end{enumerate}