A solution to the Pompeiu problem

Abstract

Let $f \in L_{loc}^1 (\R^n)\cap \mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of distributions, and $$\int_{\sigma (D)} f(x) dx = 0 \quad \forall \sigma \in G, \qquad (*)$$ where $D\subset \R^n$ is a bounded domain, the closure $\bar{D}$ of which is diffeomorphic to a closed ball, and $S$ is its boundary. Then the comp$ is connected and path connected. By $G$ the group of all rigid motions of $\R^n$ is denoted. This group consists of all translations and rotations. A proof of the following theorem is given. Theorem 1. {\it Assume that $n=2$, $f\not\equiv 0$, and (*) holds. Then $D$ is a ball.} Corollary. {\it If the problem $(\nabla^2+k^2)u=0$ in $D$, $u_N|_S=0$, $u|_S=const\neq 0$ has a solution, then $D$ is a ball.} Here $N$ is the outer unit normal to $S$.