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≥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 L2 with respect to the surface measure has ± 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 2×2 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.
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BibTeX
@misc{colbrook2026counterexample,
title = {{A counterexample to Kenig's conjecture for the Laplace double-layer operator}},
author = {Matthew J. Colbrook and Siavash Sadeghi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.07947},
eprint = {2609.07947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07947},
}
RIS
TY - UNPB
TI - A counterexample to Kenig's conjecture for the Laplace double-layer operator
AU - Matthew J. Colbrook
AU - Siavash Sadeghi
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.07947
UR - https://arxiv.org/abs/2609.07947
AB - We disprove Kenig's 1994 spectral-radius conjecture for the double-layer operator in every dimension d≥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 ± 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 2×2 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.
ER -