Math 6410 • ODE

Sobolev proved an imbedding inequality of the form
\(\norm{f}_B\le M\norm{f}_A\), where \(A=H^m(G)\) and \(B\) is the set of functions \(u\) such
that \(D^\alpha u\) is uniformly continuous on \(G\) for \(\lvert\alpha\rvert\le k\). Give,
without proof, the conditions on the open set \(G\subset\R^n\) and the integers \(m\) and
\(k\).

Select with a check-mark \checkedchoice{} and solve either B-4.I or B-4.II:

\choicebox{} \textbf{B-4.I:} Under Sobolev's conditions on \(G\), \(m\) and \(k\), each
\(f\in H^m(G)\) satisfies \(\partial^\alpha f=D^\alpha g\) a.e. for some \(k\)-times
continuously differentiable function \(g\), \(\lvert\alpha\rvert\le k\).
\begin{enumerate}
\item[(a)] Prove this, assuming the imbedding inequality above.
\item[(b)] Determine for \(n=2\) the least \(m\) such that an element in \(H^m(G)\) has 4
continuous derivatives (\(G\) as above).
\end{enumerate}

\choicebox{} \textbf{B-4.II:} Regularity theory implies that certain abstract boundary
value problems \(a(u,v)=(F,v)\) can be solved for \(u\in H^{2+s}(G)\) provided
\(F\in H^s(G)\). Consider the Dirichlet problem for \(\Delta u=F\), \(x\in G\).
\begin{enumerate}
\item[(a)] Assume \(G\) is a disk in \(\R^2\). Give without proof a smoothness condition on
\(F\) for the existence of \(C^2\) solutions \(u\).
\item[(b)] Assume \(G\) is bounded and open in \(\R^2\). Give without proof conditions on
\(G\) and \(\partial G\), and a smoothness condition on \(F\), for the existence of \(C^2\)
solutions \(u\).
\end{enumerate}