  This paper considers overdetermined boundary problems. Firstly, we give a
proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth
order in the two dimensional case and under an additional condition for the
case of dimension no less than three. Secondly, we prove an integral identity
for an overdetermined problem of fourth order which can be used to deduce
Bennett's symmetry theorem. Finally, we prove a symmetry result for an
overdetermined problem of second order by integral identities.
