A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian

[2608.27535] A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Skip to main content Search Submit Donate Log in Mathematics > Spectral Theory arXiv:2608.27535 (math) [Submitted on 27 Aug 2026] Title: A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Authors: Zikang Deng View PDF HTML (experimental) Abstract: A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian Zikang Deng Levitin and Yagudin conjectured in Conjecture 4.3 of LMS J. Comput. Math. 6 (2003) that, if the third Dirichlet eigenvalue of a connected planar domain is simple, then not all nodal lines of a corresponding eigenfunction can be closed. We construct a counterexample: a bounded connected planar domain with smooth boundary whose third Dirichlet eigenvalue is simple and for which the entire interior nodal set of every corresponding nonzero real eigenfunction is compactly contained in the domain. In fact, the nodal set consists of exactly two disjoint real-analytic simple closed curves. Starting from the Payne-conjecture counterexample of Dahne, Gomez-Serrano, and Hou, we connect two reflected copies by a symmetric thin neck. Reflection splits the spectrum into mixed Dirichlet-Neumann and Dirichlet spectra on a half-domain. A transverse Poincare inequality and the min-max principle yield convergence of both spectra to the base-domain spectrum, while boundary unique continuation gives strict separation at equal indices and places the even second branch at the third spectral position. Local eigenfunction convergence preserves strict sign inequalities; Courant's theorem confines the nodal set; and an Euler count determines its topology. A smooth inner exhaustion completes the construction. Subjects: Spectral Theory (math.SP) ; Analysis of PDEs (math.AP) Cite as: arXiv:2608.27535 [math.SP] (or arXiv:2608.27535v1 [math.SP] for this version) https://doi.org/10.48550/arXiv.2608.27535 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Zikang Deng [ view email ] [v1] Thu, 27 Aug 2026 15:47:48 UTC (16 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.SP < prev | next > new | recent | 2026-08 Change to browse by: math math.AP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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