A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures

Abstract

The planar Pompeiu problem, originating in 1929, and the associated Schiffer conjecture are long-standing rigidity questions linking rigid-motion integral transforms and Fourier zero sets to overdetermined Neumann eigenvalue problems. We construct a bounded simply connected noncircular domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $(\hat{I}”+k^2)u=0$ in $\hat{I}\text{c◯}$, $u=1,{\partial }_{\hat{I}}½u=0$ on $\partial \hat{I}\text{c◯}$ for some $k\in (31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\hat{I}\text{c◯}$ is a counterexample to Schiffer's conjecture. Green's identity also gives $\widehat{1_{\hat{I}}\text{c◯}}(k\ddot{I}‰)=0$ $(\ddot{I}‰\in {\mathbb{S}}^1)$, so $\hat{I}\text{c◯}$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains. We obtain the domain as $\hat{I}\text{c◯}=\ddot{I}•(\mathbb{D})$, where $\ddot{I}•$ is a ten-fold symmetric conformal map close to an explicitly listed polynomial of degree $301$. On the unit disc, the analytic problem becomes a cubic operator equation on real coefficient spaces, $F(g,p)=g+|p{|}^2(1+Kg)=0$, where $K$, expressed in a disk-polynomial basis, is an explicit inverse of the Laplacian on the range compatible with zero Dirichlet and Neumann traces, and $p=k\ddot{I}{•}^{\prime }$. Positivity of the disk-polynomial linearisation coefficients, sharp bounds for $K$, and monotone control of the infinite tails establish an a posteriori contraction near the listed polynomial in a weighted coefficient algebra, and hence an exact zero of $F$.