## On overdetermined boundary value problems for the biharmonic operator.(English)Zbl 0810.35020

Summary: 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.

### MSC:

 35J40 Boundary value problems for higher-order elliptic equations 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions 58J70 Invariance and symmetry properties for PDEs on manifolds
Full Text: