# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.03032