Symmetry in an overdetermined fourth order elliptic boundary value problem.(English)Zbl 0612.35039

The author modifies an elementary argument used by H. F. Weinberger [Arch. Ration. Mech. Anal. 43, 319-320 (1971; Zbl 0222.31008)] for a second order overdetermined problem to deduce that if $$u\in C^ 4({\bar \Omega})$$ is a solution of the overdetermined fourth order problem $$\Delta^ 2u=-1$$ in $$\Omega \subset R^ N$$, $$u=\partial u/\partial n=0$$ on $$\partial \Omega$$, and $$\Delta u=c$$ (constant) on $$\partial \Omega$$, then $$\Omega$$ is a ball of radius $$[| c| (N^ 2+2N)]^{1/2}$$ and u is radially symmetric in $$\Omega$$. A characterization of open balls by means of an integral identity is also presented.
Reviewer: P.Schaefer

MSC:

 35J40 Boundary value problems for higher-order elliptic equations 35C05 Solutions to PDEs in closed form 35A30 Geometric theory, characteristics, transformations in context of PDEs

Zbl 0222.31008
Full Text: