Overdetermined boundary value problems for the biharmonic operator

In the early seventies, Serrin studied an overdetermined boundary value problem and he determined that if a solution exists, then the domain must be a ball. Since then many authors have extended his results to other overdetermined boundary value problems. In this work we give a detailed development of the proof of the basic overdetermined problem first considered by Serrin as done by Weinberger using an elementary argument. We then consider five overdetermined boundary value problems for the equation Δ²u = f(r) in Ω. Two auxiliary conditions are given on interior surfaces and the extra condition is given on the boundary. We show that Ω must be a ball and, moreover, determine an integral representation of the solution for these problems.