On overdetermined problems for the homogeneous and inhomogeneous biharmonic equation (Q1379601)

The paper concerns the overdetermined boundary value problem \(\Delta^2u=1\) in a bounded domain \(\Omega\subset \mathbb{R}^n\), \(u=b\), \(|\nabla u|=c\) and \(\Delta u=d\) on the boundary \(\partial\Omega\), where \(b\), \(c\), and \(d\) are constants satisfying suitable restrictions. It is proved that if the above problem has a solution, then \(\Omega\) must be a ball. The method used is reducing the given problem to an equivalent overdetermined second order problem and then using known results of Serrin and of \textit{G. A. Philippin} and \textit{L. Ragoub} [Z. Angew. Math. Phys. 46, No. 2, 188-197 (1995; Zbl 0826.35029)].