A counterexample to Kenig's conjecture for the Laplace double-layer operator

Abstract

We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension $d\ge 2$. For every $t>1/2$ sufficiently close to $1/2$, we construct a bounded simply connected Lipschitz domain with connected boundary such that the double-layer operator on $L^2$ with respect to the surface measure has $\pm \mathrm{i}t$ in its Fredholm essential spectrum. In particular, the essential spectral radius is greater than $1/2$. In the plane, the boundary may be chosen differentiable everywhere and smooth away from a single point. Our planar construction develops a refinement mechanism that produces unbounded growth of adjoint resolvent solutions with fixed forcing while keeping the graph slopes uniformly bounded. A closed recursion for the density and its image under an auxiliary singular integral reduces the growth estimate to a certified inequality for Hermitian $2\times 2$ matrices. The certificate uses continuous estimates, exact rational arithmetic and rigorous interval enclosures. Localisation and rescaled graph insertions then produce a weakly null sequence of normalised approximate eigenvectors for the adjoint double-layer operator on one bounded boundary. A compact intertwining relation on Cartesian products of the planar domain with cubes gives the higher-dimensional examples without changing the spectral parameter.