Math 6410 • ODE

Suppose that \(u(\mathbf{x})\) 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 \textit{mean value property}: if \(\mathbf{x}\in\Omega\) and
\(r>0\) is chosen such that \(B_r(\mathbf{x})\subset\Omega\) (ball of radius \(r\)
centered at \(\mathbf{x}\)) then
\[
  u(\mathbf{x})=\frac{1}{\omega_n r^{n-1}}\int_{\partial B_r(\mathbf{x})}u(s)\dd s,
\]
where \(\omega_n\) is the measure of \(\partial B_1\). Hence show that
\[
  u(\mathbf{x})\le\frac{n}{\omega_n r^n}\int_{B_r(\mathbf{x})}u(\mathbf{y})\dd y.
\]
\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}