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

Solution Description

This work constructs a bounded simply connected noncircular domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary and a nonconstant function $u$ such that $$ (\Delta+k^2)u=0\quad\hbox{in }\Omega, \qquad u=1,\qquad \partial_\nu u=0 \quad\hbox{on }\partial\Omega $$ for some $k\in(31.967007261,31.967007293)$. Thus $u$ is a Neumann eigenfunction which is constant on the boundary, and $\Omega$ is a counterexample to Schiffer's conjecture. Green's identity gives $$ \widehat{\mathbf 1_\Omega}(k\omega)=0 \qquad(\omega\in\mathbb S^1), $$ so $\Omega$ fails the Pompeiu property and is also a counterexample to the planar Pompeiu conjecture for bounded simply connected Lipschitz domains.

The domain is parametrized as $\Omega=\phi(\mathbb{D})$, where $\phi$ 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 $p=k\phi'$. 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 $F$.

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

   • Matthew J. Colbrook,
   • George Stepaniants

   arxiv.org, 2026arXiv