Math 6410 • ODE

Suppose that \(u(x_1,x_2)\) is a \(C^2\) harmonic function in the domain
\(\Omega\subset\R^2\), so \(\Delta u=0\) in \(\Omega\).
\begin{enumerate}
\item[(a)] Prove the \textit{mean value property}: if \(y=(y_1,y_2)\in\Omega\) and
\(r>0\) is chosen such that \(B_r(y)\) (the ball of radius \(r\), centered at \(y\)) satisfies
\(B_r(y)\subset\Omega\), then
\[
  u(y_1,y_2)=\frac{1}{2\pi}\int_0^{2\pi}
  u(y_1+r\cos\theta,y_2+r\sin\theta)\dd\theta.
\]
\item[(b)] Assuming \(\Omega\) is connected, prove that \(u\) can attain a maximum value
at an interior point \(x\in\Omega\), only if \(u\) is constant.
\end{enumerate}