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 \(B_r(y)\subset\Omega\) (ball of
radius \(r\) centered at \(y\)) 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 its maximum value
at an interior point \(x\in\Omega\), only if \(u\) is constant.
\end{enumerate}