A computer-assisted counterexample to the planar Berenstein conjecture

Abstract

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $\Omega$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in (27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^{\omega }(\overline{\Omega })$ satisfying $(\Delta +k^2)u=0$ in $\Omega$, with $u=0,{\partial }_{\nu }u=\text{constant}\ne 0$ on $\partial \Omega$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{{\sigma }_{\partial \Omega }}(k\omega )=0$ for $\omega \in {\mathbb{S}}^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

24 pages, 1 figure