A note on the frequency gaps between integers in the thin obstacle problem

Federico Franceschini; Ovidiu Savin

REVIEW 2 major objections 5 minor 1 cited by

A note on the frequency gaps between integers in the thin obstacle problem

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that homogeneous solutions to the thin obstacle problem in any dimension cannot have frequencies in the intervals $(2k,2k+1)$, so in particular the interval $(2,3)$ is forbidden.

desk verdict A short, credible proof excluding frequency intervals (2k,2k+1) in the thin obstacle problem; the main theorem is sound, the only soft spot is a too-terse proof of Lemma 2. read the letter →

arxiv 2412.13425 v1 pith:HZYNCIHZ submitted 2024-12-18 math.AP

Federico Franceschini , Ovidiu Savin This is my paper · ORCID

classification math.AP MSC 35R3535B40

keywords thinobstacleproblemhomogeneoussolutionsfrequencygapsfreeboundaryregularityradialcomparisonsolutionsphericalLaplacianintegrationbypartsblow-upfrequencies

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 thin obstacle problem describes a surface that can touch a lower-dimensional obstacle, and near contact points its solutions look like homogeneous functions with a frequency parameter $\lambda$. This note proves that, in every dimension $n\ge 1$, no homogeneous solution has a frequency belonging to any interval $(2k,2k+1)$, $k\in\mathbb{N}$; in particular the interval $(2,3)$ is entirely forbidden. The argument is a short integration by parts on the upper half-sphere, comparing an arbitrary solution with a specially chosen radial solution. The result sharply narrows the spectrum of possible frequencies and removes entire open intervals from the set of admissible blow-up rates.

What carries the argument

The central object is the radial solution $p_\lambda$, defined on the upper half-sphere $S_+$ by the ODE (3): $p''+(n-2)\cot\phi\,p'+\lambda(\lambda+n-2)p=0$ with $p(0)=1$, $p'(0)=0$. This function is the $\lambda$-homogeneous harmonic extension of the value $1$ at the pole. Lemma 2 records that at the equator $\phi=\pi/2$ the signs of $p_\lambda$ and $p'_\lambda$ match $\cos(\lambda\pi/2)$ and $-\sin(\lambda\pi/2)$; the proof tracks how zeros and critical points of $p_\lambda$ are created one at a time as $\lambda$ passes each integer. The integration-by-parts identity (4) then converts those signs into a contradiction for $\lambda\in(2k,2k+1)$. The proof of Lemma 2 relies on the monotonicity of the first eigenvalue of a spherical cap together with a Sturm comparison argument to count zeros.

What would settle it

Compute $p_\lambda$ and $p'_\lambda$ at $\phi=\pi/2$ numerically, say for $n=3$ and $\lambda=2.5$, by solving the ODE (3); if the signs are not opposite, Lemma 2 is false and the proof collapses. A more direct falsification would be any explicit nonzero homogeneous thin obstacle solution with frequency $\lambda\in(2,3)$, which would contradict Theorem 1.

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Extended reading notes

Core claim

Theorem 1 states that for any $\lambda$-homogeneous solution to the thin obstacle problem in $\mathbb{R}^n$ with $\lambda>0$ and $n\ge 1$, $\lambda$ cannot lie in the union of intervals $(2k,2k+1)$. The proof constructs the radial function $p_\lambda$ that is harmonic, homogeneous of degree $\lambda$, and equals $1$ at $e_n$; Lemma 2 gives the sign relations that the signs of $p_\lambda(\pi/2)$ and $p'_\lambda(\pi/2)$ are $\cos(\lambda\pi/2)$ and $-\sin(\lambda\pi/2)$, respectively. For $\lambda$ in $(2k,2k+1)$ these signs are opposite, so the integration-by-parts identity (4) forces both boundary integrals to vanish. That would make $u$ a simultaneous Dirichlet and Neumann eigenfunction on the half-sphere, hence zero, contradicting $\lambda>0$.

Load-bearing premise

The load-bearing premise is the zero-counting step in Lemma 2: as $\lambda$ increases through a positive integer, the number of zeros of $p_\lambda$ in $[0,\pi/2]$ increases by exactly one, and not more; if this count could jump by two or skip a zero, the sign pattern used to produce the contradiction would fail.

Editorial extensions

If this is right

Every homogeneous thin obstacle solution in any dimension $n\ge 1$ has frequency outside the union of intervals $(2k,2k+1)$; in particular $(2,3)$ contains no frequency.
The intervals $(2k+1,2k+2)$ remain the only open candidates for non-integer frequencies between consecutive integers, which is consistent with the known constructions of non-2D frequencies in the authors' previous work.
For free-boundary classification, any blow-up limit at a free-boundary point cannot have such frequencies, so the possible leading-order behaviours of solutions are restricted.
The one-page proof gives a simple necessary condition on $\lambda$ that may be checked directly from the ODE for $p_\lambda$ in other settings.

Reading between the lines

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

The same sign argument might transfer to other free boundary problems where a radial comparison function with oscillatory sign exists; this would be a natural extension the paper does not make.
The zero-counting step suggests a concrete conjecture: the frequency set $\Lambda$ may be contained in the union of integers, half-integers of the form $2k+\tfrac32$, and the intervals $(2k+1,2k+2)$; Theorem 1 rules out the complementary intervals but does not prove containment.
One can test Lemma 2 numerically for moderate $k$ by solving the ODE (3) and checking that the signs at $\phi=\pi/2$ agree with $\cos(\lambda\pi/2)$ and $-\sin(\lambda\pi/2)$; failure would pinpoint a gap in the proof, not necessarily a counterexample to the theorem.

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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

Colleague,

Here's my read on Franceschini-Savin, arXiv:2412.13425. The short version: it proves a genuinely new exclusion result—no homogeneous thin-obstacle solution can have frequency in any interval (2k,2k+1), in any dimension—and does it with a clean integration-by-parts argument. The paper is honest and self-contained; the new step is the sign observation for the radial solution p_λ and its derivative at the equator, and the resulting contradiction with the sign conditions of the obstacle problem.

What's good: Theorem 1 is new, as far as I can tell. Prior work classified Λ∩(0,2) and gave examples at integers and half-integers, but the gap intervals are untouched. The proof is short and elegant. Identity (4) is correct, and the sign contradiction is clear once Lemma 2 is granted. The comparison argument in Lemma 2 is standard Sturm-Liouville lore, and the lemma itself is true—it is essentially the endpoint behavior of the Gegenbauer solution. No circularity; the self-citation to [FS] is only for motivation, not as a premise.

Where I have a quibble: the proof of Lemma 2 is compressed exactly where the stress-test note points. The claim that the number of zeros of p_λ in [0,π/2] increases by exactly one when λ crosses an integer does not follow from the stated comparison alone. The argument gives a lower bound on the zero count (a zero of p_{λ'} between consecutive zeros of p_λ), but it does not by itself rule out two zeros entering at interior points, nor does it give the upper bound needed for the exact count. The concluding sentence "increases exactly by one" is asserted rather than derived. In practice the lemma is correct—one can prove it by representing p_λ in terms of Legendre functions or by a standard node-counting theorem for Sturm-Liouville problems with monotone weight—but the note as written leaves a real gap for a referee to fill. This is a small-to-moderate exposition issue, not a fatal flaw. It should be fixable in a few lines.

The citation pattern looks fine. No data, no fitting, no invented entities. The paper doesn't address the complementary intervals (2k+1,2k+2), but it doesn't claim to.

Net: this is a solid within-subfield contribution, suitable for a serious journal. I'd send it to review, with a request that the authors expand the proof of Lemma 2. I'd cite it if I worked on thin obstacle frequencies.

Regards.

Referee Report

2 major / 5 minor

Summary. This short note proves that the frequency set Λ of the thin obstacle problem contains no frequencies in the intervals (2k,2k+1) for k∈N, in all dimensions. The proof is based on an integration-by-parts identity on the upper half-sphere S+ that relates the boundary integrals of u and its normal derivative to the endpoint values of a radial λ-homogeneous harmonic function pλ. The key auxiliary result, Lemma 2, asserts that pλ(π/2) and pλ′(π/2) have the signs of cos(λπ/2) and −sin(λπ/2), respectively, which yields the sign contradiction when λ lies in one of the forbidden intervals. The main theorem then follows by ruling out the case where both boundary integrals vanish. The paper is self-contained apart from the cited literature, and the companion paper [FS] is used only for motivation.

Significance. If the argument is completed, the result is a clean and general-dimensional restriction on the frequency set of the thin obstacle problem, settling in particular the question for the interval (2,3). The integration-by-parts identity is elegant and would give a short proof of a previously unknown gap structure. However, the proof as written does not fully establish Lemma 2, which is load-bearing: the endpoint signs in equation (4) are obtained from a zero-counting assertion that is not proved in sufficient detail. The gap appears repairable by standard Sturm-Liouville or explicit Gegenbauer-function techniques, so the result is plausible even though the manuscript is not yet complete.

major comments (2)

[§2, Lemma 2 and its proof] The proof of Lemma 2 rests on the assertion that the number of zeros and critical points of pλ in [0,π/2] remains constant between consecutive integers and increases by exactly one as λ crosses each integer. The preceding argument only establishes a monotonicity and interlacing statement: between two consecutive zeros of pλ there must be at least one zero of p_{λ′} for λ′>λ. This gives a lower bound on the zero count for larger λ, but no upper bound. It does not rule out two zeros entering at interior points within the same λ-interval, and it does not force the count to change by exactly one at integer values of λ. The sentence invoking monotonicity of the first eigenvalue of a spherical cap is not expanded and does not by itself supply the missing exact-count statement. Since the sign contradiction in equation (4) is derived from the endpoint signs, the proof of Theorem 1 hinges on this unproved step. Please provide a complete proof of the zero-counting claim, or equivalently prove Lemma 2 directly from the explicit representation of pλ as a Gegenbauer/Jacobi function and evaluate the endpoint values using connection formulas.
[§2, same paragraph] The Sturm comparison assertion that between any two consecutive zeros of pλ there must be at least a zero of p_{λ′} is not immediate from the ODE as written because the equation (3) contains a first-order term (n−2)cotφ·p′ and a singular coefficient at φ=0. A rigorous comparison argument would first reduce the equation to normal form by the change of variables q(φ)=sin^{(n−2)/2}(φ)p(φ), then apply the standard Sturm comparison theorem on a compact subinterval away from the singularity. This is a missing technical detail in the proof of Lemma 2. It is likely fixable, but as written the comparison step is asserted rather than demonstrated.

minor comments (5)

[Abstract] There are typographical errors in the abstract: 'obst acle' should be 'obstacle', 'there re are' should be 'there are', and 'homogenous' should be 'homogeneous'.
[§1] In the introduction, the phrase 'the the thin obstacle problem' contains a duplicated article; it should read 'the thin obstacle problem'.
[§1, equation (2)] The symbol N is used in (2) without definition; presumably it denotes the positive integers, and this should be stated explicitly.
[§2, proof of Lemma 2] The notation in the comparison paragraph is ambiguous: pλ′ is used both for the derivative of pλ and for the solution at a different parameter λ′. Please use p_{λ′} for the latter to avoid confusion.
[References] The reference [FS] is listed as 'In preparation' and is used only for motivation. It should be clearly stated that the main theorem does not depend on the results of [FS], to avoid any appearance of circularity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from the ODE and Sturm comparison, and the self-citation [FS] is only motivational.

full rationale

The proof of Theorem 1 is self-contained in its load-bearing steps: it reduces the existence of a nonzero homogeneous solution to the boundary identity (4), and then uses the sign information on p_lambda and p'_lambda at phi=pi/2 from Lemma 2. Lemma 2 is derived from the ODE (3) by Sturm comparison, eigenvalue monotonicity, and continuity in lambda, not from the target frequency-gap theorem or from any fitted data. The self-citation [FS] appears only in the introduction as motivation and context; it is not used as a premise in the proof of Theorem 1. The zero-counting paragraph in Lemma 2 is concise and may be a rigor gap, because it asserts that the number of zeros and critical points increases exactly by one at each integer without fully proving the upper bound, but a possible proof gap is not circularity: it does not assume the theorem being proved, and it does not rename an input as a prediction. No definitional identity, fitted-input prediction, or load-bearing self-citation chain is present.

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

The proof relies only on standard comparison principles, maximum principles, and the standard regularity framework for the thin obstacle problem. There are no fitted parameters, ad hoc constants, or new postulated entities; the auxiliary function pλ is a classical harmonic homogeneous solution.

assumptions (4)

standard math Sturm comparison theorem for the second-order ODE (3)
Used in the proof of Lemma 2 to argue zeros of pλ interlace as λ increases; not proved in the paper.
standard math Strong maximum principle for elliptic equations
Invoked in the contradiction argument in Lemma 2 and in the final step to conclude u≡0 from vanishing Dirichlet and Neumann data.
standard math Monotonicity of the first eigenvalue of a spherical cap
Used in Lemma 2 to ensure p_{λ'} vanishes before the first zero of pλ; stated without proof.
domain assumption Existence and C^{1,1/2} regularity of λ-homogeneous thin obstacle solutions
The proof uses the boundary sign conditions (1) and the equation Lλ u=0 on S\Z(u), which follow from the standard formulation of the thin obstacle problem.

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Pith. "Pith review of A note on the frequency gaps between integers in the thin obstacle problem." pith.science (2026). https://pith.science/paper/HZYNCIHZ

@misc{pith2026241213425,
  author       = {Pith},
  title        = {Pith review of: A note on the frequency gaps between integers in the thin obstacle problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZYNCIHZ}},
  note         = {Machine review of arXiv:2412.13425}
}

abstract

We give a simple proof of the fact that - in all dimensions - there are no homogeneous solutions to the thin obstacle problem with frequency $\lambda$ belonging to intervals of the form $(2k,2k+1)$, $k \in \mathbb{N}$. In particular, there are no frequencies in the interval $(2,3)$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
math.AP 2024-12 conditional novelty 8.0 of 10

For almost every obstacle, the degenerate part of the free boundary in the fractional obstacle problem vanishes up to dimension 3 for every s in (0,1).

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · cited by 1 Pith paper

[1]

The structure of the free boundary for lower dimensional obstacle problems

Athanasopoulos, I.; Caffrelli, L.A.; Salsa, S. The structure of the free boundary for lower dimensional obstacle problems. Amer. J. Math. 130 (2008), no. 2, 485-498

work page 2008
[2]

Direct epiperimetric inequalities for the thin obstacle problem and applications

Colombo, M.; Spolaor, L.; Velichkov, B. Direct epiperimetric inequalities for the thin obstacle problem and applications. Comm. Pure Appl. Math. 73 (2020), no. 2, 384-420

work page 2020
[3]

Generic regularity of free boundaries for the obstacle problem

Figalli, A.; Ros-Oton, X.; Serra, J. Generic regularity of free boundaries for the obstacle problem. Publ. Math. Inst. Hautes \'Etudes Sci. 132 (2020), 181-292

work page 2020
[4]

Franceschini F., Savin O.; Solutions to the thin obstacle problem with non 2D frequencies In preparation

work page
[5]

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

Garofalo, N.; Petrosyan, A. Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415-461

work page 2009
[6]

Savin O., Yu H., Contact points with integer frequencies in the thin obstacle problem. Comm. Pure Appl. Math. 76 (2023), no. 12, 4048--4074

work page 2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

The structure of the free boundary for lower dimensional obstacle problems

Athanasopoulos, I.; Caffrelli, L.A.; Salsa, S. The structure of the free boundary for lower dimensional obstacle problems. Amer. J. Math. 130 (2008), no. 2, 485-498

work page 2008

[2] [2]

Direct epiperimetric inequalities for the thin obstacle problem and applications

Colombo, M.; Spolaor, L.; Velichkov, B. Direct epiperimetric inequalities for the thin obstacle problem and applications. Comm. Pure Appl. Math. 73 (2020), no. 2, 384-420

work page 2020

[3] [3]

Generic regularity of free boundaries for the obstacle problem

Figalli, A.; Ros-Oton, X.; Serra, J. Generic regularity of free boundaries for the obstacle problem. Publ. Math. Inst. Hautes \'Etudes Sci. 132 (2020), 181-292

work page 2020

[4] [4]

Franceschini F., Savin O.; Solutions to the thin obstacle problem with non 2D frequencies In preparation

work page

[5] [5]

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

Garofalo, N.; Petrosyan, A. Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177 (2009), no. 2, 415-461

work page 2009

[6] [6]

Savin O., Yu H., Contact points with integer frequencies in the thin obstacle problem. Comm. Pure Appl. Math. 76 (2023), no. 12, 4048--4074

work page 2023