{"id":"5d74b7c6-648b-4019-b631-2da6ccf96e21","arxiv_id":"2509.00455","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every integer m≥4 there are smooth non-disk domains in R^2 on which Δu+λu=0 admits a solution with constant nonzero Dirichlet and Neumann data, the first such counterexamples to the Willms-Gladwell conjecture.","lead":"This note constructs smooth, non-circular domains in the plane where an overdetermined Helmholtz boundary value problem has a solution, contradicting a 1994 conjecture of Willms and Gladwell. The construction uses local bifurcation from the unit disk and gives explicit asymptotic formulas for the domains and solutions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1.1's numerical verification of Bessel inequalities is the only load-bearing gap; if W1,4(j0,2) is not rigorously negative, Theorem 1.2 for m=4 is unproved.","rationale":"The reader identified the same essential gap: the final numerical inequality W1,4(j0,2) < 0 is load-bearing for the interval I and hence for the bifurcation argument. My independent check of the surrounding proof found no other serious defect: the Crandall–Rabinowitz application is standard, the Fredholm and kernel computations are consistent, and the reconstruction from the abstract branch to the original free-boundary problem is sound. There is a minor notational issue in the reflection symmetry condition (2.9b), where for a complexified vector field one expects w(\\bar z) = \\overline{w(z)} rather than w(\\bar z) = w(z); the proof of Lemma 3.2 effectively uses the correct conjugation, so this does not undermine the construction. I also note that the proof uses a second numerical comparison, j4,1 > j1,2, but this is of the same nature and is also certifiable. The result is very likely correct, and the appropriate disposition remains conditional pending a rigorous interval proof or an explicit statement that these inequalities are numerically verified assumptions. Since the reader's verdict already captures this, I recommend no change.","tokens_in":12137,"tokens_out":12854,"duration_ms":142210,"concrete_test":"Use a validated interval-arithmetic package (e.g., Arb or MPFI) to compute certified enclosures of j1,2, j4,1, j0,2 and W1,4(j0,2) = J1(j0,2)J4'(j0,2) − J4(j0,2)J1'(j0,2), recalling J0(j0,2) = 0. Verify j4,1 − j1,2 > 0 and W1,4(j0,2) < −10^(-12) with rigorous error bounds. If both hold, the only non-rigorous step in Lemma 1.1 is certified; if W1,4(j0,2) is not negative, Theorem 1.2 must be re-examined for m = 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.2 depends on Lemma 1.1, specifically on µm ∈ (j1,1, j0,2) for m = 4 (monotonicity then covers m > 4). Section 4 proves existence, simplicity, and monotonicity of µm, but the upper bound µm < j0,2 rests on two uncertified numerical assertions: (i) j4,1 ≈ 7.5883 > j1,2 ≈ 7.0156, used to force Jm > 0 on (0, j1,2) and hence establish the root µm in (j1,1, j1,2); and (ii) W1,4(j0,2) ≈ −0.012148 < 0, used to conclude µ4 < j0,2. These sign conditions imply J0(µm) < 0, J1(µm) < 0, Jm(µm) > 0, making the denominators in (1.4) nonzero and selecting c(0) < 0. The CAS value in (ii) is quoted without error bounds, reproducibility, or an interval proof. If W1,4(j0,2) were actually positive, then µ4 would lie outside the interval I = (j1,1, j0,2) on which the linear analysis is set up, and the existence of the m = 4 branch would not be shown. Thus the headline claim for all m ≥ 4 stands or falls on this numerical check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12538,"tokens_out":3984,"duration_ms":46673,"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":[{"comment":"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":"Section 4, proof of Lemma 1.1, final paragraph"},{"comment":"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.","section":"Section 4, inequality (4.3)"}],"minor_comments":[{"comment":"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":"Section 1.1, Remark 1.6"},{"comment":"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.","section":"Section 3, proof of Theorem 1.2"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central construction appears sound, and the only serious obstacle is the unproved numerical certification of W1,4(j0,2) < 0 and, secondarily, the zero comparisons underlying (4.3). Both are fixable within the scope of the paper—for instance by a short interval-arithmetic appendix—so this is not a rejection. I would urge the editor to require that fix before publication, since the paper's main theorem is directly conditional on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nA 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.\n\nWho 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.","headline":"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.","tokens_in":12939,"tokens_out":2645,"would_cite":true,"duration_ms":28547,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B32","35J05","35N25","35R35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["overdetermined Helmholtz problem","Willms–Gladwell conjecture","bifurcation from a simple eigenvalue","Bessel functions","Wronskian roots","conformal mapping","m-fold symmetry","sign-changing solutions"],"falsifier":"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.","tokens_in":12065,"feed_emoji":"📐","tokens_out":8437,"duration_ms":96545,"temperature":0.7,"pith_summary":"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.","feed_headline":"First non-disk domains solve overdetermined Helmholtz problem","feed_subtitle":"Near the unit disk, m-fold symmetric counterexamples appear for every m≥4, disproving the planar conjecture.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"States the conjecture being disproved and supplies the saddle-point theorem used to guarantee each constructed solution has a saddle point.","marker":"[WG94]"},{"why":"Originates the simple-eigenvalue bifurcation theorem on which the construction relies.","marker":"[CR71]"},{"why":"Supplies the real-analytic Crandall–Rabinowitz theorem (Theorem 8.3.1) used to produce the bifurcating curve.","marker":"[BT03]"},{"why":"Provides the Wronskian and interlacing identities used to prove Lemma 1.1 on the roots µm.","marker":"[P´ a13]"},{"why":"Shows the unit ball is an isolated solution except for a discrete set Λ, locating where the new bifurcation points lie and framing the semilinear interpretation.","marker":"[CR08]"}],"fun_headline_variants":["Non-disk domains break Helmholtz overdetermined conjecture","First non-disk solutions to overdetermined Helmholtz","Helmholtz overdetermined: non-disk domains exist","Disproving Willms-Gladwell: non-disk Helmholtz solutions","Non-circular domains solve Helmholtz overdetermined"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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