On Overdetermined Boundary Value Problems for the Biharmonic Operator

We consider several overdetermined boundary value problems for the biharmonic operator and deduce that if a solution exists, then the domain must be a ball in N-space. When the auxiliary conditions are prescribed on the boundary, we reduce the problem to an equivalent second order overdetermined problem and use Rellich and Green identities and/or maximum principle arguments. When one or more auxiliary condition is specified on an interior surface, we use a particular solution of an associated problem to obtain the result.