Solution to the Pompeiu problem and the related symmetry problem

Abstract

Assume that $D\subset \mathbb{R}^3$ is a bounded domain with $C^1-$smooth boundary. Our result is: {\bf Theorem 1.} {\em If $D$ has $P-$property, then $D$ is a ball.} Four equivalent formulations of the Pompeiu problem are discussed. A domain $D$ has $P-$property if there exists an $f\neq 0$, $f\in L^1_{loc}(\mathbb{R}^3)$ such that $\int_{D}f(gx+y)dx=0$ for all $y\in \mathbb{R}^3$ and all $g\in SO(2)$, where $ SO(2)$ is the rotation group. The result obtained concerning the related symmetry problem is: {\bf Theorem 2.} {\em If $(\nabla^2 +k^2)u=0$ in $D$, $u|_S=1$, $u_N|_S=0$, and $k>0$ is a constant, then $D$ is a ball.}