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 $\Delta u+k^{2}u=0$ 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 ${\int }_S{e}^{ik\beta \cdot s}ds=0,\forall \beta \in S^2$ implies that $S$ is a sphere.

the argument is not complete