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New Homogeneous Solutions for the One-Phase Free Boundary Problem

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Pith's one-line read For every large integer m, the paper constructs a domain on S^2 with m+2 boundary components whose first Laplace eigenvalue is exactly 2 and whose boundary satisfies the constant-gradient condition of the one-phase free boundary problem, br

desk verdict Serious construction paper: the 3D theorem is detailed and likely correct, the 4D theorem is deferred and needs more detail, but this deserves peer review. read the letter →

arxiv 2509.09409 v1 pith:4THZ7ZY3 submitted 2025-09-11 math.AP math.DG

Coleman Hines , James Kolesar , Peter McGrath This is my paper
classification math.APmath.DG MSC 35R3535J0558J5049Q10
keywords one-phasefreeboundaryproblemhomogeneoussolutionsextremaldomainsLaplaceeigenvaluesminimalsurfaceslinearizeddoublingLDoverdeterminedellipticproblems
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, for every sufficiently large integer m, a domain on the round 2-sphere with m+2 boundary components whose first Laplace eigenvalue equals 2 and whose boundary satisfies the constant-normal-derivative condition of the one-phase free boundary problem. The domains have only finite dihedral symmetry, unlike all previously known examples in three dimensions, which were rotationally symmetric. If correct, this answers an open question and disproves a conjecture that only two symmetric examples exist. The same method yields an infinite family in dimension four with m^2 boundary components.

What carries the argument

The key mechanism is the LD solution φ = τ_0 Φ_0 + τ_2 Φ_2, a symmetric eigenfunction with logarithmic singularities at m equator points and two poles, whose singularity strengths are fixed by the vanishing-mismatch equations. The paper proves a bounded right inverse for the shifted Dirichlet-to-Neumann operator B on the oscillation subspace of boundary functions; this operator controls the dominant linear term in the normal derivative of the eigenfunction under domain perturbations, allowing a Schauder fixed point to close the nonlinear iteration uniformly in m.

What would settle it

For a specific moderately large m (say m=12), solve the linearized-doubling equation numerically to high precision, compute the mismatch vector M_Lφ, and check whether the radii τ_0, τ_2 given by equations (3.9)-(3.10) actually make both components vanish to the predicted order O(m^2 τ_0^3). A failure of the mismatch to vanish would invalidate the starting configuration.

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Extended reading notes

Core claim

The central discovery is that minimal-surface 'doublings' of the equator in S^3 carry canonical solutions to the overdetermined eigenvalue problem when pulled back to the sphere. Specifically, the authors show that linearized-doubling (LD) solutions — singular eigenfunctions with logarithmic blow-up at prescribed points — provide approximate solutions whose mismatch can be tuned to zero, and that a perturbative scheme based on a shifted Dirichlet-to-Neumann operator turns these into exact solutions on domains that are small geodesic disks removed from S^2. The resulting domains are invariant under D_m x Z_2, have m+2 real-analytic boundary components, and are not rotationally symmetric.

Load-bearing premise

The entire iteration depends on the bounded right inverse for the shifted Dirichlet-to-Neumann operator B having operator norm independent of the discrete parameter m; if that norm grew with m, the error terms would overwhelm the linear control and the fixed point would not close.

Editorial extensions

If this is right

  • There exist infinitely many non-rotationally-symmetric extremal domains for the first Laplace eigenvalue on S^2, contradicting the previously conjectured classification.
  • The domains provide homogeneous solutions to the one-phase free boundary problem in R^3 with finite symmetry groups and arbitrarily many free-boundary components.
  • The same construction works in S^3, producing domains with m^2 boundary components and giving an infinite family of homogeneous solutions in R^4.
  • The boundary components are not constant-mean-curvature spheres, disproving a related conjecture about the shape of extremal boundary components.
  • The examples are necessarily unstable for the Alt-Caffarelli functional, so they probe the class of non-minimizing free-boundary solutions where few explicit examples are known.

Reading between the lines

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

  • The connection to minimal doublings suggests a general dictionary: any family of LD solutions with tunable singularities could yield new extremal domains in higher dimensions or other space forms.
  • Since the construction leaves one free parameter after scaling, there may be a one-parameter family of solutions for each large m, not just isolated domains.
  • The method likely extends to other eigenvalue constraints by shifting the linear operator accordingly, potentially producing extremal domains for higher eigenvalues.
  • A numerical implementation for moderate m (say 10–20) could test the predicted scaling τ ~ e^{-√m/2} and verify λ_1 = 2 to high precision, providing a concrete check of the construction.
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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 constructs, for each sufficiently large integer m, a domain Ω_m⊂S^2 with m+2 real-analytic boundary components, invariant under D_m×Z_2, that admits a solution to the overdetermined problem (1.4), i.e. an extremal domain for the first Laplace eigenvalue with eigenvalue 2. The construction starts from Kapouleas-type LD solutions with singular set L_0∪L_2, solves the matching equations to obtain an approximate solution φ, develops an m-uniform linear theory for the shifted Dirichlet-to-Neumann operator on S^2\D, and closes a Schauder fixed point. The same scheme is claimed in dimension four, producing domains in S^3 with m^2 boundary components. The paper thereby answers a question of Jerison–Kamburov and disproves a conjecture of Souam and a related conjecture of Hong.

Significance. If the proofs are correct, Theorem 1.1 is a substantial result: it provides the first non-rotationally symmetric extremal domains in S^2 with finite isometry group, resolving an open question and a conjecture. The method connecting minimal-surface doublings to the one-phase free boundary problem is new and likely to be influential. The proof of Theorem 1.1 is largely self-contained and contains many explicit m-uniform estimates; in particular, the boundedness of the right inverse in Proposition 4.13 is supported by a detailed construction, contrary to the concern that it might hide m-dependence. The main weaknesses are an apparent error in the displayed matching equation (3.11), which is load-bearing for the asymptotic sizes of τ_0 and τ_2, and the abbreviated proof of the dimension-four theorem in Section 6.

major comments (2)
  1. [§3.4, Eq. (3.11)] As printed, equation (3.11) cannot be the correct matching equation. Substituting r=√(m/2)−(1/4)log m, the regime claimed in (i)(b), gives m/2−r^2≈(1/2)√(m/2)log m while the logarithmic term is ≈−(3/2)√(m/2)log m, so the left-hand side is ≈−√(m/2)log m, not zero. Deriving from (3.9)–(3.10) yields instead m/2−r^2+r log(e r/(2m e^{Φ'_0(p0)}))=0, or an equivalent form. The subsequent expression for ζ in (3.12) is consistent with the corrected equation but not with the printed one. Since the sizes of τ_0 and τ_2 feed into every later estimate, this must be corrected and the cancellation leading to (3.12) displayed.
  2. [§6 / Theorem 6.28] Theorem 1.2 is a central advertised result, but its proof is not written: the paper says it is 'essentially identical to that of Theorem 5.11', and several lemmas in Section 6 are deferred with 'obvious notational changes' (e.g. Lemma 6.18(iv), Proposition 6.19 Step 2, Proposition 6.20). The fixed-point argument is the heart of the construction; please provide the dimension-four analogue of Theorem 5.11's proof, or at least a precise point-by-point dictionary that verifies all constants are m-independent and that the operator B in (6.22) satisfies the same estimates used to close the Schauder argument.
minor comments (4)
  1. [§2.2, Lemma 2.6] The notation D_{2m} for a dihedral group of order 2m is nonstandard and conflicts with the use of D_m in Theorem 1.1. Please harmonize the notation (e.g. use D_m for the dihedral group of order 2m and G_m≅D_m×Z_2).
  2. [§4.2, Prop. 4.13 Step 2] Reference typo: '(6.23)' should be '(4.23)'. Similarly, in Prop. 5.10 the reference to '(6.31)' should be '(5.12)'.
  3. [§6] Several occurrences of S^2 should be S^3 in the dimension-four section: Prop. 6.19(i), Corollary 6.23(i), and Lemma 6.18's domain notation should be checked.
  4. [Introduction, §1 after Thm 1.1] The assertion that 'there is no solution of (1.4) arising in this way' for the simpler equator-only candidate is stated without proof or reference. Either supply a proof or mark it as a heuristic/observation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the construction solves genuine matching equations and a fixed-point problem; reliance on prior LD solutions is an external dependency, not an assumed conclusion.

full rationale

The paper's central claim (Theorem 1.1) is an existence result for λ1-extremal domains in S^2 with m+2 boundary components. The derivation does not assume this conclusion. The approximate solution φ is obtained in Lemma 3.12 by solving the mismatch equations M_iφ = 0, which are nonlinear equations determining the radii τ0,τ2; these are not fitted parameters renamed as predictions. The final solution is obtained in Theorem 5.11 by a Schauder fixed point applied to the map N(w), with the operator B carrying the dominant linear term; Proposition 4.13 constructs the right inverse R via R = R̃(BR̃)^{-1} after proving ∥BR̃−I∥ ≤ C/√m in (4.24). This is a standard perturbation argument, not a definitional identity. The dependencies on Kapouleas/McGrath LD solutions [28,29,30] and Kapouleas–Zou [31] supply linear singular solutions of the Jacobi equation and are not equivalent to the extremal-domain conclusion. Some of these references are self-citations by the authors, but they are used for background linear elliptic facts (existence/uniqueness of LD solutions, estimates) that do not contain the theorem being proved. The skeptical concern about uniformity in m of the right-inverse bound is a genuine technical condition, but it is a correctness/verification issue, not circularity. Section 6 omits proofs for Theorem 6.28 with 'we don't repeat the details'; this is an omitted-detail concern, not circular reasoning. Overall, no circular step is exhibited; the proof is a genuine gluing/perturbation construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The construction imports constants and objects from earlier work (LD solutions, Green functions, ODE profiles) but does not fit parameters to data or postulate new physical entities. The radii τ_i are solved via the matching equations, not chosen ad hoc, and no new particles, forces, or dimensions are introduced.

assumptions (5)
  • domain assumption Existence and uniqueness of G_m-symmetric LD solutions with prescribed configuration (Lemma 3.2)
    Cited from Kapouleas [28, Lemma 3.10]; used to define the building block φ=τ0Φ0+τ2Φ2.
  • standard math Green's function estimates for L=Δ+2 on S^2 (Lemma 2.10) and for L=Δ+3 on S^3 (Lemma 6.7)
    Quoted from [28, Lemma 2.20] and [31, Lemma 4.1]; underpin the asymptotic expansions of the LD solutions.
  • standard math ODE facts for rotationally invariant solutions on S^3 (Lemma 6.9)
    Quoted from [31, Lemma 2.7]; used in the dimension-four construction.
  • standard math Schauder fixed point theorem and standard elliptic regularity / Fredholm alternative
    Used in Theorem 5.11 and in constructing H_L, J_L and the right inverse.
  • domain assumption Symmetry group G_m acts transitively on L_0 and L_2, forcing the eigenspace decomposition in Lemma 2.8
    The proof of the spectral gap for B depends on these symmetries.

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Pith. "Pith review of New Homogeneous Solutions for the One-Phase Free Boundary Problem." pith.science (2026). https://pith.science/paper/4THZ7ZY3

@misc{pith2026250909409,
  author       = {Pith},
  title        = {Pith review of: New Homogeneous Solutions for the One-Phase Free Boundary Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4THZ7ZY3}},
  note         = {Machine review of arXiv:2509.09409}
}
abstract

For each sufficiently large integer $k$, we construct a domain in the round $2$-sphere with $k$ boundary components which is the link of a cone in $\mathbb{R}^3$ admitting a homogeneous solution to the one-phase free boundary problem. This answers a question of Jerison-Kamburov, and also disproves a conjecture of Souam left open in earlier work. The method exploits a new connection with minimal surfaces, which we also use to construct an infinite family of homogeneous solutions in dimension four.

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