Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

Miles H. Wheeler

REVIEW 2 major objections 3 minor 3 cited by

Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane

T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For each m≥4, smooth bounded planar domains that are not disks admit solutions of the overdetermined Helmholtz problem with constant Dirichlet and Neumann data, giving the first counterexamples to the Willms–Gladwell conjecture.

desk verdict A clean bifurcation construction that almost certainly refutes the Willms–Gladwell conjecture in the plane; one uncertified numerical inequality in Lemma 1.1 is the only barrier to full rigor. read the letter →

arxiv 2509.00455 v1 pith:7ZTEDB7E submitted 2025-08-30 math.AP

Miles H. Wheeler This is my paper · ORCID

classification math.AP MSC 35B3235J0535N2535R35

keywords overdeterminedHelmholtzproblemWillms–GladwellconjecturebifurcationfromasimpleeigenvalueBesselfunctionsWronskianrootsconformalmappingm-foldsymmetrysign-changingsolutions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper constructs smooth bounded domains in the plane, other than disks, on which the Helmholtz equation has a nonconstant solution with a constant value and a constant normal derivative on the boundary. Such domains were conjectured never to exist, and only partial positive results were known. The construction bifurcates off the unit disk: for every integer m≥4 there is a one-parameter family of m-fold symmetric domains and solutions, with explicit leading-order Bessel-function expansions. If the construction is correct, the Willms–Gladwell conjecture is false in two dimensions, and the produced solutions are real-analytic and sign-changing.

What carries the argument

Crandall–Rabinowitz bifurcation from a simple eigenvalue, applied to a reformulation of the free-boundary problem in the fixed unit disk. The reformulation uses conformal mappings together with a change of dependent variables that partially decouples the linearized equations. The bifurcation points are the positive roots µm of the Wronskian W1,m(µ)=J1(µ)Jm′(µ)−Jm(µ)J1′(µ), which are shown to be simple, decreasing in m, and contained in (j1,1,j0,2). The kernel of the linearized operator at µm is spanned by an explicit pair constructed from Bessel functions, and transversality is verified by differentiating a Wronskian identity.

What would settle it

Evaluate W1,4(j0,2)=J1(j0,2)J4′(j0,2)−J4(j0,2)J1′(j0,2) with rigorous interval arithmetic; the theorem requires this value to be strictly negative. Independently, for m=4 numerically solve the bifurcation equation near the predicted Wronskian root and small ε to check that a one-parameter family of non-disk domains actually exists; if either check fails, the central claim collapses.

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Extended reading notes

Core claim

The paper's central discovery is that the overdetermined Helmholtz problem has smooth, bounded, non-disk solutions in the plane. Specifically, for every integer m≥4 there is an ε0>0 and a curve of classical solutions (u(ε),Ω(ε),c(ε),λ(ε)) for ε∈(−ε0,ε0) with b=1, where Ω(ε) is m-fold symmetric and is not a disk for ε≠0. The domain is the image of the unit disk under a conformal map with leading correction ε z^{m+1}, and the solution has an explicit leading-order expansion in Bessel functions. The branch bifurcates from the radial solution at µ=µm, the first positive root of the Wronskian W1,m=J1Jm′−JmJ1′, and the proof rests on Lemma 1.1 showing that such roots exist, are simple, decrease wi

Load-bearing premise

The proof's final step is a single numerical inequality—W1,4(j0,2)≈−0.012148<0—checked only with a computer algebra system; if that sign is wrong, the bound µm<j0,2 and the sign choices in the leading-order expansions fail, so the branch is not shown to exist.

Editorial extensions

If this is right

The Willms–Gladwell conjecture is false in R²: there exist smooth bounded simply connected domains, not disks, for which the overdetermined Helmholtz problem has a solution with nonzero constants b and c.
For each m≥4 a separate m-fold symmetric branch exists, so non-disk solutions come in infinitely many distinct symmetry types, including arbitrarily high rotational order.
The solutions are real-analytic up to the boundary and sign-changing; reinterpreted as an overdetermined semilinear problem, they give a simpler family of sign-changing solutions for the nonlinearity f(u)=u+b than earlier constructions.
By the paper's scaling symmetry, the normalized value b=1 can be replaced by any nonzero constant, so the non-disk phenomenon is not special to one Dirichlet data value.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same Wronskian mechanism should produce analogous branches in higher dimensions, with cos mθ replaced by spherical harmonics and Jm by the corresponding radial eigenfunction; verifying this would extend the planar counterexamples to the conjecture in Rⁿ.
A rigorous interval-arithmetic check that W1,4(j0,2)<0 would remove the only non-rigorous step in the proof, making the theorem fully computer-independent.
Because µm decreases with m and is bounded below by j1,1, the bifurcation points have a limiting value as m→∞; exploring the limiting configuration could clarify whether a degenerate bifurcation occurs at the first zero of J1.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a good paper. It gives the first smooth bounded simply connected domains in R^2, other than disks, for which the overdetermined Helmholtz problem has a solution with all constants λ, b, c nonzero. That directly contradicts the 30-year-old Willms–Gladwell conjecture, and the construction via local bifurcation from the unit disk is both clever and clearly explained. The change of variables that decouples the linearized operator is the key move; it makes the kernel computation reduce to a Bessel Wronskian problem that is handled analytically for the most part.

The proof is a textbook Crandall–Rabinowitz application. The Banach spaces are set up carefully, the Fredholm and transversality checks are explicit, and the asymptotics in Theorem 1.2 are internally consistent. I checked the sign bookkeeping in Section 3 and it holds up. The paper is also honest about what is new: prior counterexamples lived on spheres, annuli, unbounded domains, or used different nonlinearities, so the claim of first simply connected planar examples is accurate.

The soft spot is exactly what the stress-test note says, and the reader's report is right to make the verdict conditional. Lemma 1.1 establishes the interval j1,1 < μm < j0,2 that is load-bearing: it fixes the signs of J0(μm) and J1(μm), guarantees the denominators in the leading-order expansion are nonzero, and selects the sign of c. The lower bound and monotonicity are proven analytically. The upper bound μm < j0,2 for m = 4, however, rests on the numerical assertion W1,4(j0,2) ≈ −0.012148 < 0, quoted without error bounds or certification. If that inequality were false, the m = 4 branch would not be shown to exist, and the theorem for all m ≥ 4 would fall. The value is not particularly close to zero, so my prior is that the inequality is true and the paper's conclusion is very likely correct. But as written, the proof of Lemma 1.1 has a genuine gap.

A second numerical input, j4,1 ≈ 7.5883 > j1,2, is used to force Jm > 0 on (0, j1,2). That one is less concerning because these Bessel zero values are classical and the inequality is standard; a citation would close the loop.

Who benefits: anyone working on overdetermined elliptic problems, symmetry theory, or the Pompeiu/Schiffer circle. This is a short, readable note that answers a longstanding question in the plane. A serious referee should see it. My recommendation: send it to review, and ask the author to either prove the W1,4 inequality rigorously or provide an interval-certified computation. That is a small fix for a result that looks right.

Referee Report

2 major / 3 minor

Summary. The paper constructs smooth bounded domains in the plane, other than disks, on which the overdetermined Helmholtz problem (1.1) admits nonconstant solutions with constant nonzero boundary values b and c. The main result, Theorem 1.2, asserts that for every integer m ≥ 4 there is a real-analytic curve of m-fold symmetric solutions bifurcating from the unit disk, with explicit leading-order expansions (1.4). The proof reformulates the problem in a fixed disk using conformal mappings and a change of variables that partially decouples the linearized system, then applies a real-analytic Crandall–Rabinowitz theorem. The bifurcation points are roots of Bessel Wronskians W_{1,m}, analyzed in Lemma 1.1. The paper claims these are the first counterexamples to the Willms–Gladwell conjecture, which had asserted that balls are the only solutions when all constants are nonzero.

Significance. If correct, the result is significant: it disproves a long-standing conjecture in the overdetermined elliptic problem literature, complements existing local uniqueness results for the Schiffer case (c=0), and provides a simple, explicit construction in two dimensions. The approach is clean and largely self-contained, and the paper gives explicit formulas for the leading-order terms, the kernel of the linearized operator, and the transversality condition. A notable strength is that the functional-analytic setup is transparent, and most of the Bessel-function analysis is rigorous, including the monotonicity of the Wronskian roots. The main weakness is that a crucial numerical inequality in Lemma 1.1 is not rigorously established; this is load-bearing for the headline theorem.

major comments (2)

[Section 4, proof of Lemma 1.1, final paragraph] The upper bound µm < j0,2 is proved only by the statement "Using a computer algebra system we numerically calculate W1,4(j0,2) ≈ −0.012148 < 0." No error bounds, interval arithmetic, or reproducibility details are given. This sign is load-bearing: it places µ4 in I, gives J0(µm) < 0 and J1(µm) < 0, ensures the denominators in (1.4) are nonzero, and fixes the sign of c. Since monotonicity reduces m ≥ 4 to m = 4, the entire theorem stands or falls on this unchecked numerical assertion. Please replace it with a rigorous enclosure (e.g., interval arithmetic with explicit rational bounds) or an analytic proof of W1,4(j0,2) < 0.
[Section 4, inequality (4.3)] The proof that j1,2 < jm,1 for all m ≥ 4 uses the numerical values j3,1 ≈ 6.3802, j1,2 ≈ 7.0156, and j4,1 ≈ 7.5883 together with monotonicity of the zeros. The monotonicity is cited, but the strict numerical comparisons are not justified. These inequalities are needed to conclude Jm > 0 on (0, j1,2), which in turn yields the existence and simplicity of the root µm in (j1,1, j1,2). Please supply rigorous bounds or precise references for these zero inequalities, or state them as a lemma with a certified proof.

minor comments (3)

[Section 1.1, Remark 1.6] The sentence "One can also check that µm lies in the discrete set Λ defined in [CR08, Lemma 3.7 and Definition 3.9]" is unsupported and not used in the proof. Either provide a short argument or remove the comment.
[Section 3, proof of Theorem 1.2] The rescaling of ε by the factor Jm(µm)J0(µm)/(µJ1(µm)) is introduced without explicitly stating that this factor is nonzero. Since the sign conditions from Lemma 1.1 are what guarantee this, it would help the reader to state this explicitly at that point.
[Throughout] There are several typographical/rendering issues (e.g., missing spaces in the title and in displayed operators) that should be corrected in the final version. The plots in Figure 1 are so exaggerated that the shapes are qualitatively useful but should be clearly labeled as approximations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is self-contained; the only load-bearing numerical check is a correctness gap, not a circular step.

full rationale

The derivation is self-contained: Theorem 1.2 follows from the Crandall–Rabinowitz theorem applied to the explicitly constructed operator F, with the hypotheses verified in Lemmas 3.3–3.5 using classical Bessel-function identities, the monotonicity of Bessel roots, and the Wronskian root µm defined by the equation W1,m(µm)=0. No parameter is fitted to the counterexamples, no prior result by the author is invoked as load-bearing, and the leading-order expansions (1.4) are computed from the kernel eigenfunction rather than assumed. The only non-rigorous load-bearing input is the numerical check in Section 4 of Lemma 1.1: 'Using a computer algebra system we numerically calculate W1,4(j0,2) ≈ −0.012148 < 0.' This fixes the sign of J0(µm) and J1(µm) and therefore the nonzero denominators in (1.4); it is a correctness gap (the CAS value is quoted without interval bounds or reproducibility details) but not circularity, since the numerical constant is independent of the constructed solution and is not chosen to force the conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction introduces no free parameters or invented entities. The constants b, c, λ, µm are either normalized (b=1, ϕ'(0)=1), defined by the bifurcation expansion, or defined as roots of the Bessel Wronskian. The main external inputs are the Crandall-Rabinowitz theorem, standard Bessel properties, and two numerical inequalities that are used as unproved facts.

assumptions (4)

standard math Crandall-Rabinowitz real-analytic bifurcation theorem (Theorem 3.1) applies to the operator F.
The proof invokes it as a black box from [BT03] after verifying Fredholm, one-dimensional kernel, and transversality.
standard math Bessel functions solve ODE (2.6) and satisfy the Wronskian identity d/dµ(µW_{k,ℓ}) = (ℓ²-k²)/µ J_k J_ℓ (equation 4.2).
Used throughout Lemma 1.1 and Lemma 3.4 to reduce kernel and transversality conditions to root locations of Wronskians.
domain assumption Bessel root inequalities j3,1 < j1,2 < j4,1, with the stated decimal values, and monotonicity of jm,1 in m imply j1,2 < jm,1 for m≥4.
Inequality (4.3) is load-bearing for the interval I and the sign analysis; the decimal root values are accepted without proof.
ad hoc to paper W1,4(j0,2) ≈ −0.012148 < 0, from a computer algebra system.
Final step of Lemma 1.1 to ensure µm < j0,2; no proof or reproducible code is provided.

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Pith. "Pith review of Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane." pith.science (2026). https://pith.science/paper/7ZTEDB7E

@misc{pith2026250900455,
  author       = {Pith},
  title        = {Pith review of: Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZTEDB7E}},
  note         = {Machine review of arXiv:2509.00455}
}

abstract

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b &\qquad& \text{ on } \partial \Omega, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial \Omega \end{alignedat} \right. $$ has a solution for some constants $\lambda,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94].

Figures

Figures reproduced from arXiv: 2509.00455 by the authors.

**Figure 1.** Exaggerated sketches of the domains Ω(ε) and functions u(ε) from Theorem 1.2 for m = 4, 5, 6, based on the leading-order approximation (1.4). depend real-analytically on ε. (iv) As ε → 0 we have the asymptotic expansions ϕ(reiθ; ε) = reiθ + ε(reiθ) m+1 + O(ε 2 ), (1.4a) (u ◦ ϕ)(reiθ; ε) = J0(µmr) J0(µm) + εµm J1(µm)Jm(µmr) J0(µm)Jm(µm) − J1(µmr) J0(µm) r m+1 cos mθ + O(ε 2 ), (1.4b) c(ε) = −µm J1(µm) J0(µm) + O(ε… view at source ↗

**Figure 2.** Plots of J1(µ), Jm(µ), µW1,m(µ) for m = 4. Their roots satisfy the inequalities j1,1 < µm < j1,2 < jm,1, and µW1,m is strictly decreasing on the interval (j1,1, j1,2). For integers m ≥ 0, let jm,1 < jm,2 < · · · denote the positive roots of Jm, all of which are simple. Importantly, jm,n is a strictly increasing function of m for each n. From the numerical values j3,1 ≈ 6.3802, j1,2 ≈ 7.0156, and j4,1 ≈ 7.5883, we se… view at source ↗

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Forward citations

Cited by 3 Pith papers

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

M. L. Agranovsky. On the stability of the spectrum in the P ompeiu problem. J. Math. Anal. Appl. , 178(1):269--279, 1993

work page 1993

[2] [2]

Symmetry theorems related to P ompeiu's problem

Patricio Aviles. Symmetry theorems related to P ompeiu's problem. Amer. J. Math. , 108(5):1023--1036, 1986

work page 1986

[3] [3]

An inverse spectral theorem and its relation to the P ompeiu problem

Carlos Alberto Berenstein. An inverse spectral theorem and its relation to the P ompeiu problem. J. Analyse Math. , 37:128--144, 1980

work page 1980

[4] [4]

Schreiber, and B

Leon Brown, Bertram M. Schreiber, and B. Alan Taylor. Spectral synthesis and the P ompeiu problem. Ann. Inst. Fourier (Grenoble) , 23(3):125--154, 1973

1973

[5] [5]

Analytic theory of global bifurcation

Boris Buffoni and John Toland. Analytic theory of global bifurcation . Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2003. An introduction

work page 2003

[6] [6]

Berenstein and P

Carlos A. Berenstein and P. C. Yang. An inverse N eumann problem. J. Reine Angew. Math. , 382:1--21, 1987

work page 1987

[7] [7]

B. Canuto. A local symmetry result for linear elliptic problems with solutions changing sign. Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire , 28(4):551--564, 2011

work page 2011

[8] [8]

B. Canuto. Stability results for the N -dimensional S chiffer conjecture via a perturbation method. Calc. Var. Partial Differential Equations , 50(1-2):305--334, 2014

work page 2014