Proof of the Schiffer's conjecture

Abstract

The following conjecture has been known for many decades as Schiffer's symmetry problem (or Schiffer's conjecture): Assume that $\Delta u+k^2u=0$ in $D$, $u|_S=0$, $u_N|_S=1$, where $D\subset \mathbb{R}^3$ is a bounded, connected, $C^2-$smooth domain, $S$ is its boundary, $N$ is a unit normal to $S$ pointing out of $D$, $k^2>0$ is a constant. Then $S$ is a sphere. In this paper the above conjecture is proved. It is also proved that the relation $\int_Se^{ik\beta\cdot s}ds=0, \,\, \forall \beta\in S^2$ implies that $S$ is a sphere.