  In two spatial dimensions, we discuss the relation between the solvability of
Schiffer's overdetermined problem and the optimality, among sets of prescribed
area, of the first eigenvalue in the buckling problem for a clamped plate and
that of the first eigenvalue of the Stokes operator. For the latter, we deduce
that the minimisers under area constraint that are smooth and simply connected
must be discs from the fact that a pressureless velocity is a necessary
condition of optimality.
