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REVIEW 4 major objections 5 minor 39 references

A computer-assisted counterexample to the planar Berenstein conjecture

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A certified construction produces a non-circular real-analytic domain whose sign-changing Dirichlet eigenfunction has constant normal derivative, disproving the unrestricted planar Berenstein conjecture.

desk verdict A credible computer-assisted refutation of the planar Berenstein conjecture; the certificate is strong, the tail estimates are mostly standard but one monotonicity claim is underexplained. read the letter →

arxiv 2608.08953 v1 pith:MRAUO4GV submitted 2026-08-09 math.AP cs.NAmath-phmath.MPmath.NAmath.SP

Matthew J. Colbrook , Siavash Sadeghi , George Stepaniants This is my paper · ORCID
classification math.APcs.NAmath-phmath.MPmath.NAmath.SP MSC 35J0535N2535P0542B1047J0565G20
keywords BerensteinconjectureoverdeterminedDirichleteigenvalueproblemconstantnormalderivativeconformalmappingdiskpolynomialsvalidatednumericscomputer-assistedproofFourierzerosets
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 claims to construct a concrete bounded, simply connected planar domain $\Omega$ with real-analytic boundary that is not a disc, together with a frequency $k\approx 27.438$ and a nonzero real-analytic solution $u$ satisfying $(\Delta+k^2)u=0$ in $\Omega$, $u=0$, and $\partial_\nu u=1$ on $\partial\Omega$. If correct, this settles the unrestricted planar Berenstein conjecture in the negative: overdetermined Dirichlet--Neumann data alone do not force a disc. The proof pulls the problem back to the unit disc through a $D_{13}$-symmetric conformal map, rewrites it as a coupled interior--boundary polynomial system, and reduces existence to finitely many certified inequalities checked by interval arithmetic. A separate sign-recovery step converts the squared boundary condition $(Ng)^2=|a|^2$ back to $Ng=|a|$, and because the eigenfunction changes sign, the classical positivity symmetry theorem does not apply.

What carries the argument

The load-bearing object is the $D_{13}$-adapted disk-polynomial basis $\Phi_{\ell,s}$, with $\Phi_{\ell,s}(r,\theta)=r^{13|\ell|}P_s^{(0,13|\ell|)}(2r^2-1)e^{i13\ell\theta}$. The positive product formula for these polynomials makes the weighted $\ell^1$ coefficient space a Banach algebra with multiplication constant one. On this algebra the paper writes down the explicit zero-Dirichlet inverse $K_D$, with separate actions on harmonic modes $s=0$ and on modes $s\ge 1$, and its Neumann trace $N$, with $N\Phi_{\ell,0}=e^{i13\ell\theta}/(2(13|\ell|+1))$ and $N\Phi_{\ell,s}=0$ for $s\ge1$. These give the coupled fixed-disc equations $F_1(g,k,a)=g+k^2|a|^2K_Dg=0$ and $F_2(g,k,a)=(Ng)^2-|a|^2=0$ on the boundary. A contraction-mapping radii-polynomial theorem reduces a zero of $F$ to finitely many inequalities, which are certified in interval arithmetic with directed rounding at 256-bit and guarded 192-bit precision.

What would settle it

Run an independent implementation of the certificate checks: if it returns $Y>3.6219\times10^{-9}$ or $Z>0.4732202748897342$ at the stated centre, or if two reproductions of the directed-rounding runs disagree, the existence theorem fails.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is Theorem 1.1: there is a bounded simply connected domain $\Omega\subset\mathbb{R}^2$ with real-analytic Jordan boundary, invariant under the dihedral group $D_{13}$ of order $26$ but neither a disc nor centrally symmetric, and a frequency $k$ in $(27.4381178838,27.4381198839)$ for which a nonzero real-valued $u\in C^\omega(\overline{\Omega})$ solves $(\Delta+k^2)u=0$ in $\Omega$, $u=0$, $\partial_\nu u=1$ on $\partial\Omega$. Equivalently, by the paper's Proposition 2.1, the arclength measure $\sigma_{\partial\Omega}$ has Fourier transform $\widehat{\sigma_{\partial\Omega}}(k\omega)=0$ for every direction $\omega\in\mathbb{S}^1$. The function $u$ changes sign. The construction is not a formal adaptation of the companion Neumann-endpoint framework: the nonzero Neumann datum keeps the harmonic source modes present, so the problem becomes the coupled system $g+k^2|a|^2K_Dg=0$ and $(Ng)^2=|a|^2$ on the boundary, together with a separate sign-recovery argument.

Load-bearing premise

Everything rests on the correctness of the authors' own computer verification: the bounds $Y\le 3.6219\times 10^{-9}$, $Z\le 0.4732202748897342$, and the contraction checks were produced by their own interval-arithmetic code and aggregated by their own checker, with no independent formal verification reported.

Editorial extensions

If this is right

  • The unrestricted planar Berenstein conjecture is false: a non-disc can carry a sign-changing Dirichlet eigenfunction with constant nonzero normal derivative.
  • There exists a non-circular real-analytic Jordan curve whose boundary arclength measure vanishes on the full Fourier circle $\{\xi:|\xi|=k\}$.
  • The overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on the eigenfunction.
  • For this counterexample the boundary obstruction is global: $|\nabla u|=1$ holds on $\partial\Omega$ even though $u$ is not sign-definite inside.
  • The construction provides a $D_{13}$-symmetric, non-centrally-symmetric example, so rigidity theorems restricted to centrally symmetric convex domains do not apply.

Reading between the lines

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

  • The construction is likely parametric: the same symmetry-adapted branch could be continued to further sign-changing examples at other frequencies and symmetry orders (editorial inference).
  • The Fourier-circle equivalence turns the problem into a search for analytic Jordan curves whose arclength transform vanishes on a whole circle, which suggests geometric searches at frequencies tied to zeros of Bessel-function combinations (editorial inference).
  • If the certified arithmetic were replaced by a formally verified implementation, the result could become a fully machine-checked theorem; the current proof depends on the authors' own directed-rounding verifier (editorial inference).
  • The sign-recovery step indicates that the distinction between the squared and unsquared Neumann condition is the true obstruction separating the Berenstein conjecture from its sign-definite versions (editorial inference).
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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

4 major / 5 minor

Summary. The paper claims a computer-assisted counterexample to the planar Berenstein conjecture in its unrestricted, sign-changing form. Theorem 1.1 asserts the existence of a bounded simply connected domain Ω in R^2 with real-analytic Jordan boundary, a frequency k in (27.4381178838, 27.4381198839), and a nonzero real-analytic function u such that (Δ + k^2)u = 0 in Ω, u = 0 and ∂_ν u = 1 on ∂Ω. The domain is D13-invariant, neither a disc nor centrally symmetric, the eigenfunction changes sign, and the boundary arclength Fourier transform vanishes on the circle |ξ| = k. The proof proceeds through a Fourier-circle equivalence (Proposition 2.1), a conformal transfer to the unit disc giving the coupled interior-boundary system (11), coefficient spaces built from D13-adapted disk polynomials, an explicit zero-Dirichlet inverse K_D and its Neumann trace N, a radii-polynomial contraction theorem (Theorem 4.1), and a computer-assisted verification of the finitely many inequalities with directed-rounding interval arithmetic. Appendix A supplies exact dyadic inputs, hashes, and a verifier script. The reconstruction part proves univalence, recovers the unsquared boundary condition by a positivity argument, extends U analytically across the boundary, and obtains the sign change via Serrin's theorem.

Significance. If the proof is correct, this is a major result: it disproves the planar Berenstein conjecture in the sign-changing case, complementing the recent Pompeiu-Schiffer counterexamples of Colbrook and Stepaniants, and it demonstrates that the fixed-disc/disk-polynomial/validated-tail framework extends to the Dirichlet endpoint with a coupled interior-boundary system. The paper is careful with normalizations, works with exact dyadic centers, and provides a detailed certificate with cryptographic hashes and a reproduction script. I found no circularity: the a posteriori contraction argument certifies existence around a fixed dyadic center rather than assuming the numerical solution. The analytic reductions, including Proposition 2.1 and the coefficient identities (18)-(20), are transparent and pass manual spot checks. The main risks are the partially unproved infinite-dimensional tail bounds and the fact that the entire numerical certificate is produced and checked by a single software pipeline.

major comments (4)
  1. [§4.4, Eqs. (45)–(47)] The shape-tail bound is load-bearing: Table 1 shows that the 'remaining shape tail, j≥61' contributes 0.4506521964853019 to Z, and Z ≤ 0.4732202748897342 is one of the two quantitative hypotheses of Theorem 4.6. The manuscript states without proof that for j≥41 the weighted-shift argument of [7, Sec. 3.5] applies because 'every angular index in H_j is positive'. For H_j = z^{mj} a° U° with U° = K_D g° expanded in the real symmetric basis Φ_{ℓ,s} + Φ_{-ℓ,s}, this positivity is not immediate and should be proved or verified from the actual support of g°. Likewise, A_- in (46) is only described as the 'exact normalised norm' after cancellation; the cancellation of the principal term of DF2 and the resulting norm should be displayed. Without these details, the infinite tail of shape directions is not independently certified.
  2. [§4.4, Eq. (41) and following paragraph] The source-tail bound is asserted to be monotone beyond the support cutoffs, but no proof is given. The extremal-column reduction to (41,0), (41,1), and (0,302) is plausible from the monotonicity of κ_{ℓ,s} = 1/[(m|ℓ|+2s)(m|ℓ|+2s+2)] and ζ_ℓ = 1/[2(mℓ+1)], but the case s = 0 with the additional trace term must be handled separately, and the present text does not do so. Since Table 1 lists the largest source-tail column meeting the retained block as 0.4593799524396485, a mistake in this step would directly affect the certified value of Z. Please include the short monotonicity argument.
  3. [§4.5, Appendix A] The theorem is certified by a single software pipeline: src/interval_assemble.cpp produces the interval enclosures, and verify_certificate.py aggregates exactly those enclosures. The verifier does not perform an independent computation of the interval bounds or of the analytic tail majorants (41), (45)–(47). Given that the main theorem rests on the certified bounds Y ≤ 3.621873700919759e-9 and Z ≤ 0.4732202748897342, I ask for a higher standard of evidence: a second independent implementation in a different language, a formal proof assistant check of the finite computations, or at minimum a complete public trace of all interval endpoints that enter Y and Z. This is a request for proportionate verification, not a demand for formal verification of the whole proof.
  4. [§3.1, §4.4, §5.3] Several essential ingredients are imported from the companion preprint [7] rather than proved in this paper: the positive disk-polynomial linearization (Corollary 3.1), the conformal transformation identities behind (10), the weighted-shift shape-tail bound (45), and the analytic-continuation argument used in Section 5.3. The paper states that some of these are independent of the symmetry order, but the manuscript should make the dependence explicit and, since [7] is a preprint, should either reproduce the arguments or give precise statements of the hypotheses verified for the present D13 data. The real-analytic regularity of u, the sign-recovery step, and the tail control all rely on these imports.
minor comments (5)
  1. [§3.1] The line 'Φℓ,s = Φ−ℓ,s' should read 'overline{Φ_{ℓ,s}} = Φ_{-ℓ,s}', and the sentence should be split so that the conjugation identity and the bound sup_D |Φ_{ℓ,s}| ≤ 1 are stated separately.
  2. [§4.4, Eq. (43) and the displayed derivative] The expression 'DF1(x°)[0,0,z^{mj}] = (k°)^2(H_j + H_j)' is ambiguous: one of the two summands should be the conjugate term \(\overline{H_j}\) arising from the second term in (23).
  3. [§4.4, Eq. (46)] The phrase 'exact normalised norm' is not defined; please state explicitly the norm and the normalization that produce A_-.
  4. [§5.2] The sentence containing '≤10−6. which together with (35)' has a typographical period before 'which'; it should read '≤10−6, which together with (35)'.
  5. [Abstract and Theorem 1.1] The abstract writes \(u \in C^\omega(\overline{\Omega})\) while the body defines \(C^\omega(\Omega)\) as restrictions of real-analytic functions defined on a neighbourhood of Ω; please unify the notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the proof is an a posteriori contraction argument using the numerical centre only as a starting point, not as the conclusion; self-citations to [7] supply general tools, not the target theorem.

full rationale

The derivation chain is not circular. The central existence result, Theorem 4.6, is a Newton–Kantorovich/radii-polynomial argument: the exact dyadic centre x° = (g°, k°, a°) is an approximate solution, but the theorem does not assume F(x°) = 0. Instead, the residual bound Y ≤ 3.6e-9 and derivative defect Z ≤ 0.473 are certified by interval arithmetic, and the radii-polynomial criterion (Theorem 4.1) produces a genuine zero x* in a ball of radius 10^-6 around x°. Thus the 'predicted' counterexample is not an input to the proof; it is the output of a contraction mapping. The Fourier moment equations (Remark 4.2) are explicitly described only as a search device and are not used as assumptions in the validation. The heavy reliance on the authors' prior preprint [7] is real but not circular in the sense defined here: the imported items — the positive disk-polynomial linearisation (Corollary 3.1), the coefficient recurrences, the weighted-shift bound (45), and the analytic-continuation scheme of Section 5.3 — are general-purpose lemmas with stated assumptions that do not include the Berenstein conjecture or the existence of the constructed domain. They are tools, not the theorem being proved. The remaining concern, that the tail majorants (41) and (45)–(47) are analytic estimates inherited from [7] and not independently checked by the verifier script, is a correctness/reproducibility risk, not a circularity: a bug in those estimates would make the proof wrong, but it would not make the proof assume its own conclusion. Similarly, the single-party nature of the directed-rounding computation is a verification-transparency issue, not a reduction of the theorem to its inputs. Accordingly, no specific circular step can be exhibited, and the overall circularity score is low.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The central claim rests on two families of inputs: (i) exact dyadic numerical centers and proof parameters tuned to make a radii-polynomial argument succeed, and (ii) a body of prior results, some classical (Rellich, Serrin, Cauchy-Kowalevski, Jacobi orthogonality) and some imported from the authors' own preprint [7] (positive linearization, conformal identities, weighted-shift tail bounds). No new physical entities are postulated. The heaviest unproven-separately input is the single-party computational certificate chain.

free parameters (2)
  • Numerical centre (k°, a°_1..a°_20, g°_ℓ,s) = k° = 3861571936906315/140737488355328 ≈ 27.43811888383842; 20 conformal coefficients and 861 g-coefficients stored as…
    Obtained by D13-symmetric branch continuation and Newton iteration near the fifth positive zero of W_{1,13} (Section 4.2). The proof treats them as exact dyadic inputs and certifies a true zero of F in a radius-1e-6 ball around them; the specific values are fitted to make F(x°) small but the existence claim does not depend on their exact digits.
  • Validation parameters (rho, sigma, tau, L, S, J_a, r) = rho = 2296835809958953/2251799813685248 ≈ 1.020000000145, sigma = 75, tau = 1, L = 20, S = 40, J_a = 20, r = 1e-6
    Chosen by hand so that the radii polynomial p(r) is negative and Z + R'(r) < 1 (eqs. (27), (38), (53)-(54)). These are ad hoc proof-tuning parameters rather than physical constants; the theorem is validated for this specific choice.
assumptions (8)
  • domain assumption Disk-polynomial positive linearization with nonnegative coefficients of total mass one ([7, Lemma 2.2])
    Imported from the authors' prior preprint [7]; gives the Banach-algebra property ||f1 f2||_rho <= ||f1||_rho ||f2||_rho used throughout Corollary 3.1 and the nonlinear majorants (29)-(31). Stated to be independent of the symmetry order m = 13.
  • domain assumption Exact conformal transformation identities [7, eq. (6)] for the Helmholtz pullback and normal-derivative scaling
    Basis for the transferred system (10) and for the identity partial_r U = |a| (partial_nu u) ∘ phi on T used in Proposition 3.5 and Section 5.2. Imported from [7].
  • domain assumption Prior inverse and weighted-shift tail results ([7, Lemma 2.4] for s >= 1 columns; [7, Sec. 3.5] for the shape-tail decay (45)-(47))
    Used in Lemmas 3.2 and 4.5 and in the shape-tail estimates; these are cited without re-derivation from the authors' own unreplicated 2026 preprint.
  • standard math Zero-Dirichlet elliptic regularity for C^2 domains, interior analytic regularity [1, 27], and the Cauchy-Kowalevski theorem [25]
    Used in Proposition 3.4 (U = K_D g is the unique W^{2,q} solution) and in Section 5.3 to extend U* real-analytically across T and obtain u* in C^omega(Omega-bar).
  • standard math Rellich uniqueness and single-layer potential trace/jump properties [8, 24]
    Used in Proposition 2.1: vanishing far field forces the exterior field to vanish, and the jump relation gamma_1^- S_k phi - gamma_1^+ S_k phi = phi produces the unit normal trace of the interior eigenfunction.
  • standard math Serrin's moving-plane rigidity for the sign-definite Dirichlet eigenfunction problem [36]
    Used twice: to infer that u* must change sign (Section 5.3) and as background explaining why any counterexample must involve a sign-changing eigenfunction (Introduction).
  • ad hoc to paper The computational verification chain (MPFR directed rounding, the FMA forward-error bound (48), and the certificate aggregator verify_certificate.py) is free of implementation error
    Theorem 4.6's existence conclusion reduces to finitely many interval enclosures. The paper pins digests and provides a verifier, but the reviewer did not re-execute the computation, and no third-party or formally verified check is reported.
  • standard math Noshiro-Warschawski univalence criterion (holomorphic map on a convex domain whose derivative has positive real part is univalent)
    Used in Section 5.1 to prove phi* is univalent on a neighborhood of D from the certified bound Re a*(z) >= 0.5727.

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Pith. "Pith review of A computer-assisted counterexample to the planar Berenstein conjecture." pith.science (2026). https://pith.science/paper/MRAUO4GV

@misc{pith2026260808953,
  author       = {Pith},
  title        = {Pith review of: A computer-assisted counterexample to the planar Berenstein conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MRAUO4GV}},
  note         = {Machine review of arXiv:2608.08953}
}
abstract

Recent work of Colbrook and Stepaniants produced the first counterexamples to the planar Pompeiu and Schiffer conjectures and introduced the conformal fixed-disc, disk-polynomial, and validated-tail machinery used here. By adapting this framework to the complementary Dirichlet endpoint, we disprove the unrestricted planar Berenstein conjecture. Specifically, we construct a bounded simply connected domain $\Omega$ with real-analytic Jordan boundary, which is not a disc and for which there exist $k\in(27.4381178838,27.4381198839)$ and a nonzero real-valued function $u\in C^\omega(\overline\Omega)$ satisfying $(\Delta+k^2)u=0$ in $\Omega$, with $u=0, \partial_\nu u=\text{constant}\ne0$ on $\partial\Omega$. Thus the overdetermined Dirichlet--Neumann data do not characterize the disc without an additional sign assumption on $u$. The domain has dihedral symmetry of order $26$, but is neither a disc nor centrally symmetric, and the corresponding eigenfunction changes sign. Equivalently, its boundary arclength measure satisfies $\widehat{\sigma_{\partial\Omega}}(k\omega)=0$ for $\omega\in\mathbb S^1$. After conformally transferring to the unit disc, exact support identities and quantitative disk-polynomial estimates yield rigorous control of the infinite-dimensional tail. A Newton--Kantorovich argument then reduces existence to finitely many explicit inequalities, which are certified using interval arithmetic. The extension from the Pompeiu--Schiffer problem is not formal. The earlier construction absorbs both boundary conditions into a single inverse-Laplacian equation. At the Dirichlet endpoint considered here, the nonzero Neumann datum forces the harmonic source modes to remain, producing a coupled interior--boundary system involving the full zero-Dirichlet inverse and its Neumann trace, together with a separate sign-recovery problem.

Figures

Figures reproduced from arXiv: 2608.08953 by the authors.

Figure 1
Figure 1. Finite-centre approximation to the sign-changing eigenfunction. The colour repre [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Works this paper leans on

39 extracted references · 25 canonical work pages

  1. [7]

    M. J. Colbrook and G. Stepaniants. A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures, 2026. arXiv:2608.01579; DOI: 10.5281/zenodo.21765317

    arXiv 2026

  2. [1]

    Agmon, A

    S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I.Communications on Pure and Applied Mathematics, 12(4):623–727, 1959

    1959

  3. [2]

    C. A. Berenstein. An inverse spectral theorem and its relation to the Pompeiu problem.Journal d’Analyse Mathématique, 37:128–144, 1980

    work page 1980

  4. [3]

    C. A. Berenstein and P. C. Yang. An overdetermined Neumann problem in the unit disk.Advances in Mathematics, 44(1):1–17, 1982

    work page 1982

  5. [4]

    C. A. Berenstein and P. C. Yang. An inverse Neumann problem.Journal für die reine und angewandte Mathematik, 382:1–21, 1987

    1987

  6. [5]

    Brown, B

    L. Brown, B. M. Schreiber, and B. A. Taylor. Spectral synthesis and the Pompeiu problem. Annales de l’Institut Fourier, 23(3):125–154, 1973

    1973

  7. [6]

    Counterexamples to Schiffer's Conjecture

    G. Cao-Labora and J. d. D. Pont. Counterexamples to schiffer’s conjecture.arXiv preprint arXiv:2608.05114, 2026

    work page Pith review arXiv 2026

  8. [8]

    Colton and R

    D. Colton and R. Kress.Inverse Acoustic and Electromagnetic Scattering Theory, volume 93 of Applied Mathematical Sciences. Springer, Cham, fourth edition, 2019

    work page 2019

Show all 39 references
  1. [9]

    G. Dai, Y. Sun, J. Wei, and Y. Zhang. On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane, 2025. arXiv:2511.19819, version 1, 25 November 2025

    2025 arXiv

  2. [10]

    Dai and Y

    G. Dai and Y. Zhang. Sign-changing solution for an overdetermined elliptic problem on unbounded domain.Journal für die reine und angewandte Mathematik, 803:267–293, 2023

    2023

  3. [11]

    Dalmasso

    R. Dalmasso. An overdetermined problem for an elliptic equation.Publications of the Research Institute for Mathematical Sciences, 46(3):591–606, 2010

    2010

  4. [12]

    Dalmasso

    R. Dalmasso. An overdetermined problem for the Helmholtz equation.Proceedings of the American Mathematical Society, 142(1):301–309, 2014

    2014

  5. [13]

    Day, J.-P

    S. Day, J.-P. Lessard, and K. Mischaikow. Validated continuation for equilibria of PDEs.SIAM Journal on Numerical Analysis, 45(4):1398–1424, 2007

    2007

  6. [14]

    Fousse, G

    L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann. MPFR: A multiple-precision binary floating-point library with correct rounding.ACM Transactions on Mathematical Software, 33(2), June 2007. Article 13, 15 pages

    2007

  7. [15]

    Garofalo and F

    N. Garofalo and F. Segala. Asymptotic expansions for a class of Fourier integrals and applications to the Pompeiu problem.Journal d’Analyse Mathématique, 56:1–28, 1991

    1991

  8. [16]

    Garofalo and F

    N. Garofalo and F. Segala. New results on the Pompeiu problem.Transactions of the American Mathematical Society, 325(1):273–286, 1991

    1991

  9. [17]

    Garofalo and F

    N. Garofalo and F. Segala. Another step toward the solution of the Pompeiu problem in the plane. Communications in Partial Differential Equations, 18(3–4):491–503, 1993

    1993

  10. [18]

    Garofalo and F

    N. Garofalo and F. Segala. Univalent functions and the Pompeiu problem.Transactions of the American Mathematical Society, 346(1):137–146, 1994

    1994

  11. [19]

    N. J. Higham.Accuracy and Stability of Numerical Algorithms, volume 80 ofOther Titles in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, second edition, 2002. 23

    2002

  12. [20]

    Hungria, J.-P

    A. Hungria, J.-P. Lessard, and J. D. Mireles James. Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach.Mathematics of Computation, 85(299):1427– 1459, 2016

    2016

  13. [21]

    IEEE Standard for Floating-Point Arithmetic

    IEEE. IEEE Standard for Floating-Point Arithmetic. IEEE Std 754-2019 (Revision of IEEE Std 754-2008), 2019

    2019

  14. [22]

    Kawohl and M

    B. Kawohl and M. Lucia. Some results related to Schiffer’s problem.Journal d’Analyse Mathéma- tique, 142(2):667–696, 2020

    2020

  15. [23]

    G. Liu. Symmetry theorems for the overdetermined eigenvalue problems.Journal of Differential Equations, 233(2):585–600, 2007

    2007

  16. [24]

    McLean.Strongly Elliptic Systems and Boundary Integral Equations

    W. McLean.Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge, 2000

    2000

  17. [25]

    R. F. Millar. The analytic continuation of solutions to elliptic boundary value problems in two independent variables.Journal of Mathematical Analysis and Applications, 76(2):498–515, 1980

    1980

  18. [26]

    I. A. Minlend. Overdetermined problems with sign-changing eigenfunctions in unbounded periodic domains, 2023. arXiv:2307.07784, version 1, 15 July 2023

    2023 arXiv

  19. [27]

    C. B. Morrey, Jr. and L. Nirenberg. On the analyticity of the solutions of linear elliptic systems of partial differential equations.Communications on Pure and Applied Mathematics, 10(2):271–290, 1957

    1957

  20. [28]

    Nigam, B

    N. Nigam, B. Siudeja, and B. Young. A proof via finite elements for Schiffer’s conjecture on a regular pentagon.Foundations of Computational Mathematics, 20(6):1475–1504, 2020

    2020

  21. [29]

    D. Pompeiu. Sur certains systèmes d’équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables.Comptes rendus hebdomadaires des séances de l’Académie des sciences, 188:1138–1139, 1929

    1929

  22. [30]

    D. Pompeiu. Sur une propriété des fonctions continues dépendant de plusieurs variables.Bulletin des sciences mathématiques (2), 53:328–332, 1929

    1929

  23. [31]

    D. Pompeiu. Sur une propriété intégrale des fonctions de deux variables réelles.Bulletin de la Classe des sciences, Académie royale de Belgique (5), 15:265–269, 1929

    1929

  24. [32]

    D. Ruiz. Nonsymmetric sign-changing solutions to overdetermined elliptic problems in bounded domains.Journal of the European Mathematical Society, 2025. Published online first, 16 January 2025

    2025

  25. [33]

    S. M. Rump. Verification methods: Rigorous results using floating-point arithmetic.Acta Numerica, 19:287–449, 2010

    2010

  26. [34]

    M. M. Schiffer. Variation of domain functionals.Bulletin of the American Mathematical Society, 60(4):303–328, 1954

    1954

  27. [35]

    M. M. Schiffer. Partial differential equations of elliptic type. In N. Aronszajn and C. B. Morrey, Jr., editors,Lecture Series of the Symposium on Partial Differential Equations, pages 97–149. Department of Mathematics, University of Kansas, Lawrence, Kansas, 1957

    1957

  28. [36]

    J. Serrin. A symmetry problem in potential theory.Archive for Rational Mechanics and Analysis, 43(4):304–318, 1971

    1971

  29. [37]

    Szegő.Orthogonal Polynomials, volume 23 ofAmerican Mathematical Society Colloquium Publications

    G. Szegő.Orthogonal Polynomials, volume 23 ofAmerican Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, fourth edition, 1975

    1975

  30. [38]

    M. H. Wheeler. Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane, 2025. arXiv:2509.00455, version 1, 30 August 2025

    2025 arXiv

  31. [39]

    S. A. Williams. A partial solution of the Pompeiu problem.Mathematische Annalen, 223(2):183– 190, 1976. 24

    1976

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Reviewed August 14, 2026 · model on record in the stance chip above.