Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen

Abstract

In dimensions $3,4,6,8,10,14$ we construct bounded convex non-ball domains with real-analytic spherical boundaries. Each admits a nonconstant solution of $Δu+u=0$ with $u=1$ and $\nabla u=0$ on the boundary. Green's identity makes their indicator Fourier transforms vanish on the unit sphere, disproving the Schiffer and Pompeiu conjectures even within the convex class in these dimensions. The three-dimensional domain is axisymmetric, invariant under $O(2)$ on the last two coordinates and reflection of the first, and is obtained by lifting a conformal map of its meridian section. In dimensions $n=2m+2$, $m=1,2,3,4,6$, the domains lie in the rank-two compact Lie algebras $\mathfrak{u}(2)$, $\mathfrak{so}(4)$, $\mathfrak{su}(3)$, $\mathfrak{so}(5)$ and $\mathfrak g_2$. Harish-Chandra's radial part formula reduces these cases to planar Helmholtz problems, and Kostant's theorem reduces convexity to the Cartan section. Exact solutions near dyadic approximations follow from computer-assisted contractions in weighted polynomial coefficient spaces, with interval bounds for finite blocks and analytic bounds for every infinite tail. A second, strictly star-shaped but nonconvex, example in dimension three is retained as an independent result. To our knowledge, apart from the earlier version of this work, these are the first counterexamples in dimensions at least three and the first known convex counterexamples in any dimension.