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

Bennett, Allan

SIAM J. Math. Anal. 17, 1354-1358 (1986).

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

Keywords:

second order overdetermined problem; fourth order; radially symmetric; characterization; integral identity

Citations:

Zbl 0222.31008