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REVIEW 2 major objections 4 minor 42 references

Counterexamples to Schiffer's Conjecture

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Pith's one-line read This paper constructs infinitely many smooth planar domains, none of them balls, that admit a Neumann eigenfunction constant on the boundary, disproving the Schiffer conjecture in the plane.

desk verdict A likely major result with a repairable but real asymptotic error in Lemma 5.2; worth sending to a serious referee. read the letter →

arxiv 2608.05114 v1 pith:LC2A434V submitted 2026-08-05 math.AP math.SP

Gonzalo Cao-Labora , Jaume de Dios Pont This is my paper · ORCID
classification math.APmath.SP MSC 35N2535B3235J0535R3533C10
keywords SchifferconjecturePompeiuproblemoverdeterminedboundaryvaluebifurcationtheoryBesselfunctionsDebyeasymptoticsDirichlet-to-Neumannmapanisotropicspaces
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

This paper claims that the Schiffer conjecture is false in the plane: there exist infinitely many smooth, bounded, simply connected, star-shaped planar domains, none of them balls, on which the Laplacian has a Neumann eigenfunction that is constant on the boundary. Because of the classical Williams equivalence, the same construction gives a negative answer to Pompeiu's problem in the plane, with the explicit test function $f(x)=e^{i\sqrt{\lambda}x_1}$. The strategy is to replace the discrete symmetry order $N$ by a real parameter $R$ and study a relaxed problem in which Bessel functions of non-integer order are allowed; bifurcation branches from such crossings are shown to have size independent of $R$, and a density result puts crossings arbitrarily close to integers. A uniform downward bend of the branch then forces it to hit an integer $R=N$, yielding genuine $N$-fold symmetric counterexamples. The authors note that their infinite family needs no computer assistance, in contrast to a concurrent computer-assisted construction with 10-fold symmetry.

What carries the argument

The load-bearing object is the relaxed functional $G(R,v):X\to Y$ defined on anisotropic Hölder spaces on a collar, whose zero set contains the trivial ball solutions $G(R,0)=0$ and whose integer-$R$ zeros unfold into $N$-fold Schiffer domains. Its linearization at a crossing $R_*$ is the Bessel-mode operator $L_R$, with kernel spanned by $\Phi_*(y)\cos\psi$, where $\Phi_*$ is built from the noninteger-order Bessel function $J_{R_*}(\sqrt{\lambda_*}(R_*-y))$; crossings are triples $(\rho,R,\lambda)$ with $J_1(\rho)=J_R(\rho)=0$ and $\lambda=\rho^2/R^2\in[2,3]$. The proof then runs on three uniform ingredients: a quantitative Crandall–Rabinowitz theorem (Proposition 3.5) that gives a branch of size independent of the crossing; Debye-type asymptotics (Lemma 5.2 and Propositions 6.1–6.2) showing that crossings accumulate near integers and that the non-resonance and symbol estimates hold uniformly; and the second-order computation $R''(0)=-\frac{\sqrt{\lambda}}{4R\,\partial_\nu j_{\nu,m}}\left(2+\rho\,J'_{2R}(\rho)/J_{2R}(\rho)\right)$, which is asymptotically a negative universal constant depending only on $\lambda$, forcing the branch to bend downward by a uniform amount.

What would settle it

Compute, for a sequence of crossings with $R_0\to\infty$, the quantities in Proposition 6.2 and the difference in (4.4): if the claimed uniform error rates fail for any crossing family with $R_*$ within $10^{-6}$ of an integer, the branch may not reach the integer and the construction collapses. Alternatively, numerically integrate the branch equations for the $N=28$ example and check whether $R(s)$ actually passes through the integer value 28 before the uniform $s$-bound is exceeded.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for infinitely many integers $N\ge 2$ there are $\lambda>0$ and a domain $\Omega\subset\mathbb{R}^2$, different from a ball, with smooth boundary, simply connected, bounded and star-shaped, such that $\Delta u+\lambda u=0$ in $\Omega$, $u=1$ and $\partial_\nu u=0$ on $\partial\Omega$, and $\Omega$ is $N$-fold symmetric, with $u$ smooth up to the boundary. The construction works by considering a collar reformulation of the problem for real symmetry $R$, with a Dirichlet-to-Neumann map encoding the interior, and proves that the functional $G(R,v)$ has nontrivial zeros when $R$ is an integer. The authors show that the linearized operator at a crossing, where $J_1(\rho)=0$, $J_R(\rho)=0$ and $\lambda=\rho^2/R^2\in[2,3]$, has a one-dimensional kernel, obtain a quantitative bifurcation branch whose length is uniform over all sufficiently large crossings, prove that crossings exist arbitrarily close to integers, and compute that the branch starts flat, $R'(0)=0$, and bends downward with $R''(0)$ uniformly bounded away from zero. Taylor's theorem then places the branch below the integer floor, so by continuity the branch value $R(s)$ takes an integer value, and Theorem 2.7 converts that zero into an actual Schiffer domain.

Load-bearing premise

The whole proof rests on the assumption that certain asymptotic formulas for Bessel functions of large real order hold uniformly over every allowed crossing family, with errors that vanish as the radii grow; if the uniformity fails, a branch might start near an integer but curve back before hitting it.

Editorial extensions

If this is right

  • If Theorem 1.1 is correct, the Schiffer conjecture is false in the plane, settling a problem from Yau's list in the negative.
  • Pompeiu's problem is also answered negatively in the plane: with $f(x)=e^{i\sqrt{\lambda}x_1}$, every rigid motion of the constructed domain integrates $f$ to zero.
  • The construction provides an infinite sequence of symmetry orders $N$, all giving smooth, bounded, star-shaped, simply connected domains, so the counterexamples are not isolated.
  • The relaxed-real-$R$ bifurcation framework transfers to other overdetermined elliptic problems in which classical perturbations are blocked by the absence of common zeros of integer-order Bessel functions.
  • The paper's main theorem and the Pompeiu corollary are accompanied by a Lean certification, so the logical steps are machine-checked.

Reading between the lines

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

  • A natural testable extension is to check whether every sufficiently large integer $N$ admits such a counterexample, not merely infinitely many; the density of crossings alone does not guarantee that every large $N$ is reached, and the uniform bend may fail for some residue classes.
  • The same collar-plus-Dirichlet-to-Neumann strategy could be adapted to other symmetry groups, such as dihedral or higher-dimensional rotational symmetries, where real-order Bessel functions would again remove the integer-order obstruction.
  • The uniform bifurcation theorem may be of independent use in free-boundary problems: it quantifies how large a branch must be to cross a discrete parameter value, which is exactly the information needed to turn non-integer formal problems into genuine solutions.
  • Because the domains have high symmetry and explicit Bessel data, one could compute the first few examples, such as the $N=28$ case shown in the paper, to high precision as a numerical check of the asymptotic constants in (4.4).
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper claims a counterexample to Schiffer's conjecture in R^2: for infinitely many integers N >= 2 there exist non-ball domains Omega, with N-fold symmetry, simply connected, bounded, star-shaped, with C^infty boundary, admitting a nontrivial solution u of (1.1) with u=1 and normal derivative 0 on the boundary. The proof reformulates the Schiffer condition on a collar through a functional G(R,v), extends the symmetry order from an integer N to a real parameter R, proves a quantitative Crandall-Rabinowitz theorem whose branch size is uniform over a family of 'crossings' (R*,rho*,lambda*), shows that crossings can be found arbitrarily close to integers via Debye asymptotics of Bessel functions, and proves that the bifurcation branch bends downward by a uniform amount so that it reaches an integer radius. Corollary 1.2 converts this into a counterexample to Pompeiu's problem.

Significance. If correct, this settles two long-standing problems in the plane and introduces a genuinely new technique: relaxing the integer symmetry order to a real parameter and proving a uniform quantitative bifurcation theorem for the resulting functional. The paper also states that the main theorem and Corollary 1.2 have been formalized in Lean, which is a strong reproducibility feature if the formalization is audited. The main unresolved technical risk is the uniformity of the Bessel-asymptotic estimates, and in particular one displayed Debye prefactor is currently wrong; this must be corrected before the proof is sound.

major comments (2)
  1. [§5.4, Eq. (5.22) and Eq. (7.15)] Lemma 5.2 is false as stated. The Debye expansion (7.3) gives J_nu(nu sec beta) = (2/(pi nu tan beta))^{1/2}(cos(xi)+O(nu^{-1})), and with nu = lambda_0^{-1/2} rho + t, nu tan beta = sqrt(rho^2 - nu^2) = rho sqrt(1 - lambda_0^{-1}) + O(1). Hence the prefactor that makes the leading cosine term O(1) is (2/(pi rho sqrt(1-lambda_0^{-1})))^{1/2}, not (pi/(2 rho sqrt(1-lambda_0^{-1})))^{1/2}. The prefactor in Lemma 5.2 is the reciprocal of the correct one up to a factor of pi/2, so the left-hand side of (5.22) decays like rho^{-1/2} instead of converging to cos(omega_0 rho - theta_0 t - pi/4). Correspondingly, Eq. (7.15) claims that the product of the two square-root prefactors is 1+O(rho^{-1}), but the actual product is of order rho^{-1}. Since Proposition 4.5 invokes Lemma 5.2 to place crossings R* arbitrarily close to integers, this is a load-bearing error. With the corrected prefactor the zero-location argument still works, because zeros of J_nu(rho) remain within O(rho^{-1}) of zeros of the cosine, which suffices for the epsilon-density argument; nevertheless Section 7 must be revised before the proof is sound.
  2. [§7, Lemmas 5.2 and 7.1, Propositions 6.1 and 6.2] The uniform Bessel-asymptotic layer is the most delicate part of the argument and is currently imported from DLMF and from Dunster's paper rather than proved in full. In particular, Proposition 4.5 requires uniformity of the Debye estimates over the crossing family C(R0), and Proposition 4.6 requires the uniform version of the order-derivative of zeros in Lemma 7.1. The prefactor error in Lemma 5.2 shows that these imported estimates need careful checking. The authors should either prove the needed uniform error bounds or quote them with precise statements, explicit non-asymptotic ranges, and verified constants, and should check that the corrected prefactors do not alter any constants used in Section 4.
minor comments (4)
  1. [Theorem 1.1] There is a typo: 'different from a a ball' should read 'different from a ball.'
  2. [§5.5 and §6] There are minor typographical errors: 'Driichlet' should be 'Dirichlet' and 'deffect' should be 'defect.'
  3. [Proposition 7.2, Eqs. (7.18)-(7.19)] The square-root notation is easy to misread: the prefactor in (7.18) is sqrt(2 pi R sqrt(4-lambda)), not sqrt(2 pi R) times sqrt(4-lambda). The text should be typeset unambiguously, since the dimensionally different reading changes the subsequent ratio computation.
  4. [Proof of Lemma 5.2] The sentence 'so that rho = nu sec beta' after defining beta = arccos(nu/rho) is correct but can be misread as an additional assumption; it would be clearer to write cos beta = nu/rho and hence rho = nu sec beta.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the construction is internally proved and its asymptotic inputs are external, parameter-free estimates.

full rationale

The paper's derivation chain is self-contained at the level of the PDE and bifurcation arguments. The counterexample is not assumed: Theorem 2.7 proves that zeros of G(R,v) with integer R unfold to Schiffer domains, and the existence of non-trivial zeros is established through the uniform Crandall–Rabinowitz theorem (Proposition 3.5), the kernel/range characterization (Lemmas 3.3–3.4), and the uniform estimates of Section 5. The crucial density of crossings near integers (Proposition 4.5) is derived from Debye-type Bessel asymptotics quoted from DLMF [30] and Dunster [16], which are external references with stated hypotheses that do not include the target theorem. The bending estimate R''(0) in Proposition 4.6 is computed in the paper via Wronskian and second-order expansion arguments, with the asymptotic formula in Proposition 7.2 following from the same external Bessel estimates. No fitted parameter is renamed as a prediction, and no author-uniqueness theorem is imported to force the conclusion. The only author-overlap citation, [7], is used descriptively as background on a similar functional setting and is not load-bearing for any step of the proof. The skeptical reviewer's concern about a prefactor in Lemma 5.2 is a correctness issue, not a circularity issue, and does not indicate that the derivation reduces to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim does not fit any data. The construction uses generic fixed constants such as the collar width L, the Holder exponent alpha, and thresholds R0 and delta, whose existence is shown rather than fitted. The mathematical assumptions are standard analytic facts plus uniform Debye-type estimates imported from DLMF and the literature. No physical entity is introduced; the real-order relaxation is a mathematical device.

assumptions (6)
  • standard math Uniform Debye-type asymptotics for Bessel functions and their order derivatives hold with stated error rates (DLMF 10.19, Dunster).
    Used in Section 7 for Lemma 5.2, Lemma 7.1, Proposition 6.2 and Proposition 7.2; these drive the near-integer crossings and the R''(0) bound.
  • domain assumption There exists lambda0 in (2,3) such that pi divided by arccos(lambda0^{-1/2}) is irrational and the phase omega0 is real.
    Proposition 4.5 uses this to make the set of Bessel zero shifts dense modulo 1. The existence is elementary but not proved in the paper.
  • standard math The classical functional-analytic toolbox applies: Schauder estimates, Fredholm theory, Gronwall, Littlewood-Paley and Besov embeddings, Privalov boundedness of the Hilbert transform.
    Used in Lemma 3.2, Proposition 4.2 and the uniform Crandall-Rabinowitz proof.
  • standard math Positive zeros of Bessel functions J_nu are simple, the first positive zero satisfies j_{nu,1} > nu for nu > 0, and consecutive oscillatory zeros are separated by a uniform constant.
    Used in Proposition 6.1 for the mode k >= 2 and k = 1 non-resonance estimates.
  • standard math Kinderlehrer-Nirenberg free boundary regularity upgrades C^{2,alpha} boundary data to real-analytic boundary and C-infinity eigenfunctions.
    Invoked at the end of Theorem 2.7 to obtain the smoothness stated in Theorem 1.1.
  • standard math Williams' reduction from Pompeiu to Schiffer is valid for simply connected domains.
    Reproduced in Section 1.2 for Corollary 1.2; the paper gives a proof, so it is not an unexamined citation.
invented entities (1)
  • Real-order symmetry parameter R, a relaxed version of the integer N
    purpose: Allows bifurcation branches to start at non-integer crossings and later reach an integer N, circumventing perturbative rigidity near the disk.
    It is a proof device defined via Bessel functions of non-integer order. It has no observable handle outside the construction; its validity is judged by the resulting integer-N solutions.

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Cite this review

Pith. "Pith review of Counterexamples to Schiffer's Conjecture." pith.science (2026). https://pith.science/paper/LC2A434V

@misc{pith2026260805114,
  author       = {Pith},
  title        = {Pith review of: Counterexamples to Schiffer's Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LC2A434V}},
  note         = {Machine review of arXiv:2608.05114}
}
abstract

The Schiffer conjecture states that if a smooth domain $\Omega \subset \mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian which is constant at the boundary, then the domain is a ball. It is intimately related to Pompeiu's problem, stating that if a nonzero function integrates zero over any rigid motion of $\Omega$, then $\Omega$ is a ball. We disprove both conjectures in $\mathbb{R}^2$, constructing infinitely many planar domains $\Omega$ which are not balls and satisfy the conditions above. Our domains are $N$-fold symmetric, with $N$ sufficiently large. Our approach is based on a novel strategy of considering a relaxed problem where $N$ can be any real number (which corresponds to the Schiffer problem only when $N$ is a natural number). We then apply bifurcation theory to this relaxed problem, showing that the size of the local bifurcation branch can be taken independently of $N$. This result allows us to conclude that branches starting with $N$ sufficiently close to an integer reach integer values of $N$.

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