OP-00711 Solution claimed

The Robin fundamental gap conjecture

For a bounded domain $\Omega \subset \mathbb{R}^d$ and $\alpha > 0$, write $\rho_1(\Omega;\alpha) \le \rho_2(\Omega;\alpha)$ for the first two eigenvalues of $-\Delta$ with the Robin condition $\partial_\nu u + \alpha u = 0$, where $\partial_\nu$ is the outward normal derivative. For an interval $(0, D)$ the same outward-normal Robin condition is imposed at both endpoints.

Among convex domains of given diameter $D$, the Neumann gap (Payne and Weinberger) and the Dirichlet gap (Andrews and Clutterbuck) are bounded below by those of the interval $(0, D)$. Andrews, Clutterbuck and Hauer proved that for $\alpha > 0$ the Robin gap of a convex domain is bounded below by $\pi^2/D^2$, the Neumann gap of $(0, D)$, whenever its Robin ground state is log-concave, and that the ground state fails to be log-concave on some convex domains. Laugesen proved the inequality below, for every real $\alpha$, when $\Omega$ is a rectangular box; there the interval is the degenerate box of diameter $D$, approached by thin boxes.

Problem. Let $d \ge 2$ and $\alpha > 0$, and let $\Omega \subset \mathbb{R}^d$ be a bounded convex domain with smooth boundary and diameter $D$. Is it true that $$\rho_2(\Omega;\alpha) - \rho_1(\Omega;\alpha) > \rho_2\bigl((0,D);\alpha\bigr) - \rho_1\bigl((0,D);\alpha\bigr)$$ for every such $d$, $\alpha$ and $\Omega$?

Source: Andrews, Clutterbuck and Hauer (2020), Section 10, as a non-strict inequality for Schrödinger operators with convex potential; Laugesen (2019), Conjecture F, with strict inequality and no potential. Stated by AIM — Open Applied Problems for smooth convex domains.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: AP
Keywords: AIM Open Applied Problems, convex domains, fundamental gap, Robin Laplacian

References

@article{laugesen2019,
  author        = {Laugesen, Richard S.},
  title         = {The {R}obin {L}aplacian---spectral conjectures, rectangular theorems},
  journal       = {J. Math. Phys.},
  volume        = {60},
  number        = {12},
  year          = {2019},
  pages         = {121507},
  doi           = {10.1063/1.5116253},
  eprint        = {1905.07658},
  archivePrefix = {arXiv},
}
@article{andrews2020,
  author        = {Andrews, Ben and Clutterbuck, Julie and Hauer, Daniel},
  title         = {Non-concavity of the {R}obin ground state},
  journal       = {Camb. J. Math.},
  volume        = {8},
  number        = {2},
  year          = {2020},
  pages         = {243--310},
  doi           = {10.4310/CJM.2020.v8.n2.a1},
  eprint        = {1711.02779},
  archivePrefix = {arXiv},
}
@article{payne1960,
  author  = {Payne, L. E. and Weinberger, H. F.},
  title   = {An optimal {P}oincar{\'e} inequality for convex domains},
  journal = {Arch. Rational Mech. Anal.},
  volume  = {5},
  number  = {1},
  year    = {1960},
  pages   = {286--292},
  doi     = {10.1007/BF00252910},
}
@article{andrews2011,
  author  = {Andrews, Ben and Clutterbuck, Julie},
  title   = {Proof of the fundamental gap conjecture},
  journal = {J. Amer. Math. Soc.},
  volume  = {24},
  number  = {3},
  year    = {2011},
  pages   = {899--916},
  doi     = {10.1090/S0894-0347-2011-00699-1},
}
@misc{aim-robin-fundamental-gap,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The Robin fundamental gap conjecture}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/652-robin-fundamental-gap.md},
  note         = {Entry 652 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Positive-Robin gap collapse in smooth convex three-dimensional domains v1 GitHub ↗

Matthew J. Colbrook

We give a candidate disproof of the diameter-normalized positive-Robin fundamental-gap conjecture stated as AIM problem \#021. A convex polytope with square end sections and one transverse central protrusion is made thin in two directions. The first transverse Robin energy has leading term equal to the boundary parameter times the fiber perimeter-to-area ratio. That ratio has two reflection-related minima and a …

Counterexample · not verified · fidelity not established · posted on GitHub 2 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: AIM @ a153f46, 021-robin-gap-counterexample/robin_gap_021.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00711, "The Robin fundamental gap conjecture", clopen.solutions.