Neumann's nodal line may be closed on doubly-connected planar domains

Pedro Freitas; Rom\'eo Leylekian

arxiv: 2604.03169 · v1 · submitted 2026-04-03 · 🧮 math.AP

Neumann's nodal line may be closed on doubly-connected planar domains

Pedro Freitas , Rom\'eo Leylekian This is my paper

Pith reviewed 2026-05-13 18:21 UTC · model grok-4.3

classification 🧮 math.AP MSC 35P1535J05

keywords Neumann eigenfunctionsnodal linesdoubly connected domainsgraph-like domainseigenvalue convergencespectral geometry

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The pith

Planar domains with one hole admit closed interior nodal lines for the first non-trivial Neumann eigenfunction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that certain doubly connected planar domains exist where the lowest non-constant Neumann eigenfunction has a nodal set forming a closed curve lying strictly inside the domain. This is optimal because Pleijel's 1956 theorem already rules out any such closed interior nodal line on simply connected planar domains. The argument proceeds by constructing thin graph-like domains whose eigenvalues and eigenfunctions converge to those of a limiting metric graph, together with a new strong transversal convergence result for the eigenfunctions. A reader cares because the result shows that adding a single hole is enough to change the possible geometry of the first nodal set.

Core claim

We show the existence of planar domains with one hole for which the first non-trivial Neumann eigenfunction has a closed nodal line fully contained inside the domain. This is optimal, as it is known since Pleijel's 1956 result that the nodal line cannot be closed on simply-connected planar domains. A part of the proof is based on the study of convergence of eigenvalues and eigenfunctions of graph-like domains towards metric graphs. We improve the known results of convergence of eigenfunctions, by showing a strong transversal convergence.

What carries the argument

Graph-like domains that converge to metric graphs, equipped with strong transversal convergence of the associated Neumann eigenfunctions.

If this is right

A single hole suffices to permit an interior closed nodal line for the first Neumann mode.
Pleijel's non-existence result is sharp once the domain is allowed to be doubly connected.
Eigenvalues and eigenfunctions on suitably thin graph-like domains converge to the spectrum of the limiting metric graph.
The new strong transversal convergence holds for the eigenfunctions under the constructions employed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Topology of the domain can control whether low-lying nodal sets close up or must hit the boundary.
The same graph-approximation technique may extend to other boundary conditions or to domains with more holes.
Direct numerical checks on specific annular domains of varying thickness would provide independent verification of the closed nodal lines.

Load-bearing premise

The thin graph-like domains must be constructed so that their eigenfunctions converge strongly in the transversal direction and thereby reproduce the desired nodal behavior.

What would settle it

A concrete sequence of doubly connected domains, each thinner than the previous, on which direct numerical solution of the Neumann eigenvalue problem shows the first non-trivial eigenfunction nodal set always reaches the boundary.

Figures

Figures reproduced from arXiv: 2604.03169 by Pedro Freitas, Rom\'eo Leylekian.

**Figure 1.** Figure 1: Schematic representation of a domain with a second Neumann eigenfunction whose nodal line (in blue) is expected to remain closed. A schematic representation of a domain for which this phenomenon occurs is shown in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Illustration of the Dirichlet FDSH constructed in [FL25]. In each case, the dotted line is the segment bisecting the rectangle and the blue line is the expected nodal line. In the Neumann case however, it is much harder to actually construct an FDSH. For instance, the Dirichlet FDSH of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Two graphs described in the introduction. To circumvent this issue, one possibility is to roll the line over itself, as in Figure 3b. Then, it becomes possible to slide the circle across the midpoint of the rolled line, without generating a second intersection. Graph-like domains obtained from Figure 3b would thus be nice candidates for a Neumann FDSH. Unfortunately they are not symmetric with respect to t… view at source ↗

**Figure 4.** Figure 4: A planar caterpillar tree with a loop. To actually locate the nodal line, one needs refined convergence results for the eigenfunctions, when the width of the graph-like domains tends to zero. As recalled in appendix A, an L 2−convergence result had already been obtained in [Pos06]. However, this is not enough to conclude that the nodal line is close to the nodal set of the limit graph. To solve this probl… view at source ↗

**Figure 5.** Figure 5: Schematic representation of the Neumann FDSH described in the introduction. The expected nodal line is drawn in blue. for the necessary and sufficient condition for the Neumann Laplacian to have compact resolvent given in [EH87] to hold. Clearly either the second eigenvalue is simple or it has a higher multiplicity. In the former case, an argument based on Courant’s nodal domain theorem and the rotational … view at source ↗

**Figure 6.** Figure 6: Unbounded domain for which there exists a first non-trivial Neumann eigenfunction whose nodal line does not touch the boundary. only two infinite arms, closer to the examples in [FK07]. However, our purpose here is only to give an idea as to how the case of unbounded domains might be handled, and we will not pursue further a more detailed analysis in this case. Open problem. We conclude the introduction by… view at source ↗

**Figure 7.** Figure 7: A caterpillar tree with a loop. The black dots are the vertices of the graph. The arrows correspond to the orientation induced by the local parametrisation (1) of the graph. This parametrisation naturally endows the edges A, B, L, R1, ..., R4 with an orientation, which is represented by the arrows in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Vertex neighborhood constructed in Step 1 of the proof of Proposition 5.1. The graph Gt is in dark pink while the neighborhood Uv,h is in light pink. Let β > 0 be the minimum of the angles between two adjacent edges and δ > 0 the minimal length of the edges in Gt. One checks that as long as tan(β/2) > 1/(αδ), which can always be achieved for some large enough α, the segments σ1, ..., σd never intersect in… view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs the first examples of doubly-connected planar domains where the first non-trivial Neumann eigenfunction has a closed nodal line strictly inside the domain.

read the letter

The main point is that the authors give explicit planar domains with one hole where the first non-trivial Neumann eigenfunction has a closed nodal line that stays away from the boundary. This is new because Pleijel showed in 1956 that no such closed line can exist on simply-connected domains, and nothing in the literature settled the doubly-connected case until now. They build the domains as thin tubes around a metric graph that contains a single cycle, then lift the isolated nodal points of the graph eigenfunction to curves in the domain via an improved transversal convergence result. The convergence strengthening is a useful technical step on its own and goes beyond the standard L2 or weak convergence statements in earlier graph approximation papers. The construction is direct and the claim is sharp in the sense that it matches the known boundary between simply- and multiply-connected cases. The soft spot is the one flagged in the stress-test note. Strong transversal convergence lifts the graph nodal points to curves, but without a uniform positive lower bound on the distance from those curves to the tube boundary, the curves could drift toward the edge as the thickness goes to zero. The abstract does not display the quantitative estimates, so the full paper must supply them to confirm the nodal line remains interior for each fixed domain. If those bounds are there and work, the argument is fine; if they are missing or only logarithmic, the existence statement weakens. This is for people working on nodal sets, Neumann eigenvalues, and metric graph approximations. Readers who follow Pleijel-type results or domain approximation techniques will get direct value from the construction and the convergence improvement. It deserves a serious referee because the existence result is clean, the method is standard but sharpened in a useful way, and the only real question is whether the interior condition is fully controlled in the estimates.

Referee Report

2 major / 1 minor

Summary. The manuscript proves existence of doubly-connected planar domains on which the first non-trivial Neumann eigenfunction has a closed nodal line lying entirely in the interior. The domains are realized as thin tubular neighborhoods of a metric graph containing one cycle; the proof proceeds by establishing an improved strong transversal convergence of the Neumann eigenfunctions on these domains to the eigenfunctions of the limiting graph, whose nodal set consists of isolated points that are lifted to closed curves inside the tube.

Significance. The result supplies the first examples in which a Neumann nodal line is closed and interior on a multiply-connected planar domain, thereby showing that Pleijel's 1956 prohibition on closed nodal lines is sharp only for simply-connected domains. The strengthened transversal convergence statement for graph-like domains is of independent interest and may apply to other thin-structure spectral problems.

major comments (2)

[Proof of strong transversal convergence (section following the graph approximation)] The argument that strong transversal convergence lifts the isolated nodal points of the graph eigenfunction to a closed curve strictly inside Ω_ε is not accompanied by a quantitative lower bound on dist(nodal set, ∂Ω_ε). Without such an estimate that remains positive for small but fixed ε (or decays slower than the tube width), the limiting nodal line could approach or intersect the boundary in every finite approximation, which would falsify the interior-containment claim.
[Domain construction and graph choice] The construction of the tubular neighborhoods must guarantee that the first non-trivial Neumann eigenfunction on the graph has a nodal set consisting of isolated points away from the vertices; the manuscript does not explicitly verify that the chosen metric graph satisfies this spectral condition uniformly in the approximation parameter.

minor comments (1)

[Notation and preliminaries] Notation for the tube radius ε and the transverse coordinate should be introduced once and used consistently; several passages switch between different symbols for the same quantity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the manuscript to incorporate the necessary clarifications and estimates.

read point-by-point responses

Referee: [Proof of strong transversal convergence (section following the graph approximation)] The argument that strong transversal convergence lifts the isolated nodal points of the graph eigenfunction to a closed curve strictly inside Ω_ε is not accompanied by a quantitative lower bound on dist(nodal set, ∂Ω_ε). Without such an estimate that remains positive for small but fixed ε (or decays slower than the tube width), the limiting nodal line could approach or intersect the boundary in every finite approximation, which would falsify the interior-containment claim.

Authors: We agree that an explicit quantitative lower bound is required to rigorously guarantee interior containment for small ε. In the revised manuscript we will derive such a bound by combining the strong transversal convergence with the isolation of the nodal points on the limiting graph (which lie at a fixed positive distance from the vertices). This yields a uniform positive lower bound on the distance to ∂Ω_ε that is independent of ε for all sufficiently small ε. revision: yes
Referee: [Domain construction and graph choice] The construction of the tubular neighborhoods must guarantee that the first non-trivial Neumann eigenfunction on the graph has a nodal set consisting of isolated points away from the vertices; the manuscript does not explicitly verify that the chosen metric graph satisfies this spectral condition uniformly in the approximation parameter.

Authors: The metric graph is fixed (a cycle with one pendant edge) and chosen precisely so that its first non-trivial eigenfunction has isolated nodal points strictly away from the vertices. In the revision we will add a short explicit verification of this spectral property, noting that it holds uniformly in the approximation parameter because the graph itself does not depend on ε. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external theorems and independent convergence improvement

full rationale

The paper constructs doubly-connected domains as thin tubular neighborhoods around a metric graph with one cycle and proves the existence of a closed interior nodal line for the first non-trivial Neumann eigenfunction by establishing strong transversal convergence of eigenfunctions to the graph eigenfunction (whose nodal set consists of isolated points). This convergence is presented as an improvement over known results rather than a reduction to prior fitted quantities or self-citations by construction. Pleijel's 1956 theorem is invoked only for the contrasting simply-connected case and is an external result. No equation or step equates the claimed nodal line property to its own inputs or to a self-citation chain; the argument remains self-contained against the stated assumptions on domain construction and convergence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard results from spectral theory on domains and metric graphs plus a new convergence statement; no free parameters or invented entities are introduced.

axioms (2)

standard math Standard spectral theory for the Neumann Laplacian on bounded domains with Lipschitz boundary
Invoked implicitly for existence of eigenfunctions and nodal set properties.
domain assumption Convergence of eigenvalues and eigenfunctions for graph-like domains to metric graphs (prior results improved here)
Central technical tool; the paper strengthens the eigenfunction convergence part.

pith-pipeline@v0.9.0 · 5371 in / 1208 out tokens · 19495 ms · 2026-05-13T18:21:17.915304+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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L. Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in ^N , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 11 (2000), 175--181

work page 2000
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Dahne, J

J. Dahne, J. Gómez-Serrano, and K. Hou, A counterexample to Payne’s nodal line conjecture with few holes, Commun. Nonlinear Sci. Numer. Simulat. 103 (2021), 105957

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Fournais, The Nodal Surface of the Second Eigenfunction of the Laplacian in ^ d Can Be Closed, J

S. Fournais, The Nodal Surface of the Second Eigenfunction of the Laplacian in ^ d Can Be Closed, J. Differential Eq. , 173 (2001), 145--159

work page 2001
[4]

Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the L aplacian on surfaces, Indiana Univ

P. Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the L aplacian on surfaces, Indiana Univ. Math. J . 51 (2002), 305--316

work page 2002
[5]

Freitas and D

P. Freitas and D. Krej c i r \' k, Location of the nodal set for thin curved tubes Indiana Univ. Math. J. 57 (2008), 343--376

work page 2008
[6]

Hoffmann-Ostenhof, T

M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, The nodal line of the second eigenfunction of the L aplacian in R ^2 can be closed, Duke Math. J. 90 (1997), 631--640

work page 1997
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Jerison, The first nodal set of a convex domain, Essays on F ourier analysis in honor of E lias M

D. Jerison, The first nodal set of a convex domain, Essays on F ourier analysis in honor of E lias M . S tein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press., Princeton, NJ, 1995, pp. 225--249

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Kennedy, Closed nodal surfaces for simply connected domains in higher dimensions, Indiana Univ

J.B. Kennedy, Closed nodal surfaces for simply connected domains in higher dimensions, Indiana Univ. Math. J. 62 (2013), 785--798

work page 2013
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Lin and W

C.-S. Lin and W. M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Am. Math. Soc. 102 (1988), 271--277

work page 1988
[10]

A. D. Melas, On the nodal line of the second eigenfunction of the L aplacian in R ^2 , J. Differential Geom. 35 (1992), 255--263

work page 1992
[11]

L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453--488

work page 1967
[12]

L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 720--729

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S. T. Yau, Open problems in geometry, in ``Proc. Symp. Pure Math.,'' 54 , Amer. Math. Soc., Providence, RI, 1993

work page 1993

[1] [1]

Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in ^N , Atti Accad

L. Damascelli, On the nodal set of the second eigenfunction of the Laplacian in symmetric domains in ^N , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 11 (2000), 175--181

work page 2000

[2] [2]

Dahne, J

J. Dahne, J. Gómez-Serrano, and K. Hou, A counterexample to Payne’s nodal line conjecture with few holes, Commun. Nonlinear Sci. Numer. Simulat. 103 (2021), 105957

work page 2021

[3] [3]

Fournais, The Nodal Surface of the Second Eigenfunction of the Laplacian in ^ d Can Be Closed, J

S. Fournais, The Nodal Surface of the Second Eigenfunction of the Laplacian in ^ d Can Be Closed, J. Differential Eq. , 173 (2001), 145--159

work page 2001

[4] [4]

Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the L aplacian on surfaces, Indiana Univ

P. Freitas, Closed nodal lines and interior hot spots of the second eigenfunction of the L aplacian on surfaces, Indiana Univ. Math. J . 51 (2002), 305--316

work page 2002

[5] [5]

Freitas and D

P. Freitas and D. Krej c i r \' k, Location of the nodal set for thin curved tubes Indiana Univ. Math. J. 57 (2008), 343--376

work page 2008

[6] [6]

Hoffmann-Ostenhof, T

M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, The nodal line of the second eigenfunction of the L aplacian in R ^2 can be closed, Duke Math. J. 90 (1997), 631--640

work page 1997

[7] [7]

Jerison, The first nodal set of a convex domain, Essays on F ourier analysis in honor of E lias M

D. Jerison, The first nodal set of a convex domain, Essays on F ourier analysis in honor of E lias M . S tein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press., Princeton, NJ, 1995, pp. 225--249

work page 1991

[8] [8]

Kennedy, Closed nodal surfaces for simply connected domains in higher dimensions, Indiana Univ

J.B. Kennedy, Closed nodal surfaces for simply connected domains in higher dimensions, Indiana Univ. Math. J. 62 (2013), 785--798

work page 2013

[9] [9]

Lin and W

C.-S. Lin and W. M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Am. Math. Soc. 102 (1988), 271--277

work page 1988

[10] [10]

A. D. Melas, On the nodal line of the second eigenfunction of the L aplacian in R ^2 , J. Differential Geom. 35 (1992), 255--263

work page 1992

[11] [11]

L. E. Payne, Isoperimetric inequalities and their applications, SIAM Rev. 9 (1967), 453--488

work page 1967

[12] [12]

L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 720--729

work page 1973

[13] [13]

S. T. Yau, Open problems in geometry, in ``Proc. Symp. Pure Math.,'' 54 , Amer. Math. Soc., Providence, RI, 1993

work page 1993