Symmetry in an Overdetermined Fourth Order Elliptic Boundary Value Problem

Let $\Omega $ be a bounded domain in $\mathbb{R}^N $ for which the following boundary value problem has a classical solution: $\Delta (\Delta u) = - 1$ in $\Omega ; = {{\partial u} / {\partial n}}$ on $\partial \Omega ;\Delta u = c$ (constant) on $\partial \Omega $. We show that $\Omega $ must be an open ball and that u must be radially symmetric about the center of $\Omega $. This result is analogous to that of Serrin (Arch. Rat. Mech. Anal., 43 (1971), pp. 304–318) and Weinberger (Arch. Rat. Mech. Anal., 43 (1971), pp. 319–320) for the problem $\Delta u = - 1$ in $\Omega ,u = 0$ and ${{\partial u} / {\partial n}} = c$ on $\partial \Omega $. Our result is obtained from a maximum principle for fourth order elliptic equations and several applications of Green’s theorem. We then obtain two characterizations of open balls by means of integral identities–the first depends on our result and the second on that of Serrin and Weinberger.