Existence and stability of solutions for a fourth order overdetermined problem

Abstract

We examine a Serrin-type overdetermined boundary value problem for the biharmonic operator. If the underlying set is the unit ball, a solution exists for a constant overdetermining condition. We prove the existence of an open and bounded domain admitting a solution to the boundary value problem for every small perturbation of the overdetermining condition. Moreover, we establish stability estimates for the deviation of this domain from the unit ball in terms of the perturbation.Our approach is motivated by a recent result of Gilsbach and Onodera and applies a result of Ferrero, Gazzola and Weth for a fourth order Steklov problem.