Math 6410 • ODE

A function \(u\in C^2(\Omega)\), \(\Omega\subset\R^n\), is subharmonic (superharmonic)
in \(\Omega\) if \(\nabla^2u\ge0\) (\(\nabla^2u\le0\)) in \(\Omega\).
\begin{enumerate}
\item[(a)] Show that if \(u\) is subharmonic then for every ball
\(B_R(\mathbf{x})\Subset\mathbf{R}\)
\[
  u(\mathbf{x})\le
  \frac{1}{\omega_n R^{n-1}}\int_{\partial B_R(\mathbf{x})}u(\sigma)\dd\sigma.
\]
Hence show that
\[
  u(\mathbf{x})\le
  \frac{n}{\omega_n R^n}\int_{B_R(\mathbf{x})}u(\mathbf{y})\dd y.
\]
How are the inequalities changed if \(u\) is superharmonic?
\item[(b)] Show that if \(u\in C(\overline{\Omega})\) is subharmonic (superharmonic),
the maximum (minimum) of \(u\) is attained only on \(\partial\Omega\), unless \(u\) is
constant.
\item[(c)] Show that if \(u\) is harmonic in \(\Omega\) then \(u^2\) is subharmonic.
\item[(d)] Let \(u\) be subharmonic in \(\Omega\) and \(F:\R\to\R\) smooth. Under what
conditions on \(F\) is \(F(u)\) subharmonic?
\end{enumerate}