Math 6410 • ODE

Assume that \(u\in C^2(\R)\), and \(f\in C_0^2(\R)\).
\begin{enumerate}
\item[(a)] Assume \(u\) satisfies \(u''=0\). Without using an explicit representation for the
solution, show that the mean value property
\[
  u(x)=\frac12\bigl(u(x+r)+u(x-r)\bigr)=\frac{1}{2r}\int_{x-r}^{x+r}u(t)\dd t,
\]
for each interval \((x-r,x+r)\), is satisfied, and conversely show that the mean value
property implies \(u''=0\).
\item[(b)] Assuming \(u''=0\) in the bounded interval \((a,b)\), use part (a) to prove the
\textit{strong maximum principle} for \(u\).
\item[(c)] Find a radially symmetric \textit{fundamental solution} \(\Phi(x)\) for the
problem \(u''=0\), and use it to prove that
\[
  u(x)=\int_{-\infty}^{\infty}\Phi(x-y)f(y)\dd y
\]
satisfies \(-u''=f\) in \(\R\).
\end{enumerate}