Math 6410 • ODE

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