Math 6410 • ODE

Suppose that \(u(x)\) is a \(C^2\) harmonic function in the domain \(\Omega\subset\R^n\),
so \(\Delta u=0\) in \(\Omega\).
\begin{enumerate}
\item[(a)] Prove the mean value property: if \(x\in\Omega\) and \(r>0\) is chosen such
that \(B_r(x)\subset\Omega\) (ball of radius \(r\) centered at \(x\)) then
\[
  u(x)=\frac{1}{\omega_nr^{n-1}}\int_{\partial B_r(x)}u(s)\dd s,
\]
where \(\omega_n\) is the measure of \(\partial B_1\). Hence show that
\[
  u(x)\le\frac{n}{\omega_nr^n}\int_{B_r(x)}u(y)\dd y.
\]
\item[(b)] Assuming \(\Omega\) is connected, prove that \(u\) can attain its maximum
value at an interior point \(x\in\Omega\), only if \(u\) is constant.
\end{enumerate}