  What can be said about the domain $\Om$ in $\bR^n$ for which its Green's
function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^\delta$? What can we
say about $\Om$ if the Boundary Harnack Principle holds in the form
$u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are
positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the
boundary also nice?
  We discuss these questions below and give answers in very special cases.
