Math 6410 • ODE

Select with a check-mark \checkedchoice{} and solve either B-3.I
or B-3.II:

\choicebox{} \textbf{B-3.I:} Let \(u,h\) denote elements of some Sobolev space and consider
the distributional differential equation \(-u''+u=h\) with Dirichlet boundary conditions
\(u(0)=u(1)=0\).
\begin{enumerate}
\item[(a)] Formulate an abstract boundary value problem \(a(u,v)=(h,v)\), by defining the
sequilinear form \(a(u,v)\), the Hilbert space \(H\) and the inner product \((\cdot,\cdot)\).
\item[(b)] Discuss in detail how the Riesz theorem applies to solve the abstract
boundary value problem.
\item[(c)] Is enough known about the Hilbert space solution \(u\) for it to be a solution
of the distributional differential equation? Explain.
\end{enumerate}

\choicebox{} \textbf{B-3.II:} Let \(\phi\in H^1_0(G)\) and denote by \(\norm{\cdot}\) the
usual norm in \(L^2(G)\), \(G\) open in \(\R^n\). Assume \(\lvert x_1\rvert\le K\) for all
\(x\in G\).
\begin{enumerate}
\item[(a)] Prove the Poincar\'e inequality
\[
  \norm{\phi}\le2K\norm{\partial_{x_1}\phi}.
\]
\item[(b)] Explain, without proof, how to use generalizations of the Poincar\'e
inequality to solve the abstract boundary value problem for distributional differential
equations of the form \(\Delta u=f\).
\end{enumerate}