Proof of the Schiffer's conjecture

TL;DR

This paper proves Schiffer's conjecture, demonstrating that certain boundary conditions imply the domain's boundary must be spherical, and establishes a related integral condition as equivalent to sphericity.

Contribution

The paper provides a rigorous proof of Schiffer's conjecture and shows the equivalence of an integral condition to the domain being a sphere.

Findings

Schiffer's conjecture is proven true.

The integral condition characterizes spherical boundaries.

The results confirm the uniqueness of spherical domains under specified boundary conditions.

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.