Some nonstandard-overdetermined boundary value problems for the biharmonic operator

In the early seventies, Serrin considered an overdetermined boundary value problem and determined that if a solution exists, then the domain must be a ball. Many authors have then extended this result to other overdetermined problems. In this work, we first examine an overdetermined problem considered by Bennett and then consider three fourth order boundary value problems for the differential equation Δ₂u = f (r) in Ω. Two of the three auxiliary conditions are assumed to hold on the surface of a ball which is completely contained in Ω. Each auxiliary condition will be of a different type. We also show that Ω must be a ball. We then determine an integral representation of the solution for these problems.