$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem

Runcao Lyu

REVIEW 3 major objections 5 minor 46 references

$C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Locally flat free boundaries are infinitely smooth for every fractional Laplacian order s in (0,1).

desk verdict Plausible C^{1,α}→C^{2,α} step for flat fractional free boundaries, but the C∞ bootstrap leans on an unverified application of [JN17] to curved slits, so the full proof needs referee attention. read the letter →

arxiv 2509.00609 v2 pith:2HVAQ6HG submitted 2025-08-30 math.AP

Runcao Lyu This is my paper · ORCID

classification math.AP MSC 35R3535B6535R11

keywords fractionalLaplacianone-phasefreeboundaryproblemregularityC∞Caffarelli–SilvestreextensionflatboundariesHarnackinequalitylinearized

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 proves that in the one-phase fractional Laplacian free boundary problem, any sufficiently flat free boundary is C∞ smooth, for every order 0

What carries the argument

The half-plane solution U(t,y)=((−t+√(t²+y²))/2)^s with free boundary L={t=y=0} is the model profile. Around it the paper builds a family V_{S,a,b} of explicit solutions whose free boundaries are paraboloids, uses domain variation maps to measure how close an arbitrary solution is to these models, and derives a quadratic improvement-of-flatness iteration. For the high-regularity step, the central object is the linearized problem L_β(u_n w)=0 with radial Neumann condition |∇_r w|=0 on the free boundary; the quotient w=u_i/u_n extends the derivatives of the boundary graph, and Whitney-extension polynomials together with boundary Harnack estimates in slit domains let the graph's C^{k,α} regular

What would settle it

Take a flat viscosity solution whose free boundary is the graph x_n=g(x') with g∈C^{1,α} but g∉C^2 near 0, constructed by prescribing suitable boundary data on ∂B1 for the extended problem. If the expansion in Theorem 2.6 fails for the linearized solution or if the quotient u_i/u_n is not C^{2,α}, Theorem 5.2 is violated; computing that quotient along a sequence of shrinking balls would settle the claim.

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Extended reading notes

Core claim

The central claim is Theorem 1.2 (proved in extended form as Theorem 1.3): there is a small ε̄ depending only on n and s such that any viscosity solution of the one-phase problem (−Δ)^s u=0 in {u>0} with the growth condition u(x+tν)/t^s→1 on the free boundary, whose zero set lies within ε̄ of a hyperplane in B1, has free boundary of class C∞ in B1/2. The proof works through the Caffarelli–Silvestre extension, turning the nonlocal problem into a degenerate local elliptic problem in one extra dimension, and establishes the result for all β=1−2s∈(−1,1).

Load-bearing premise

The C∞ bootstrap treats the C^{k,α}_{x,r} expansion estimates for u/U0 and its derivatives as a black box inherited from boundary Harnack theory in slit domains; if those estimates do not come with constants uniform through the iteration, the induction only yields finite regularity.

Editorial extensions

If this is right

Flat free boundaries for the one-phase fractional Laplacian are C∞ for every 0<s<1, not only for s=1/2.
The C∞ bootstrap passes through an explicit C^{2,α} checkpoint: flat boundaries first become C^{2,α} via paraboloid approximation, then the linearized problem upgrades them.
The free-boundary derivatives ∂_i g satisfy the same equation class as g, so the graph's regularity can be raised one derivative at a time indefinitely.
The expansion theorem for the linearized problem (Theorem 2.6) gives polynomial approximations of arbitrary order for solutions in slit domains, a tool that can be reused in nearby free boundary problems.
A follow-up analysis of graphical solutions and Bernstein-type problems is foreshadowed, since flat-boundary regularity is the key input there.

Reading between the lines

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

The same two-step scheme should extend to other degenerate elliptic operators for which a boundary Harnack principle in slit domains and a family of paraboloid model solutions exist, not only to the fractional Laplacian.
If the boundary Harnack constants in slit domains were to degrade with the iteration, the proof would yield only finite regularity; so the theorem's sharpness is tied to the uniformity of those constants, not to the nonlinear iteration itself.
A direct test of the method is to check the C^{2,α} step against explicit non-paraboloidal flat solutions, e.g. solutions with boundary data at ∂B1 chosen to force a C^{1,α} but non-C² graph; Theorem 5.2 predicts the quotient u_i/u_n must still be C^{2,α}.

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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

The honest summary: the C^{1,α}→C^{2,α} half is a real candidate contribution, but the C∞ bootstrap in Section 5 is not fully proved as written because the key input, Theorem 5.1, is deferred to a citation without checking that U0 is Lβ-harmonic when the boundary is curved.

What is genuinely new is the extension of the De Silva–Savin paraboloid iteration to the full range β ∈ (-1,1). The paper is honest that [BROW25] already covers the C^{2,α}→C∞ step, and it clearly lays out the two-step strategy. The domain variation machinery and the expansion theorem, if they work, would give a self-contained route for all s. The internal logic is coherent and I found no manufactured circularity.

The soft spot is real. In Theorem 5.1, the proof says (5.2)-(5.3) follow from [JN17, Prop. 3.2 and Lemma 7.2], but U0 is built from signed distance to a curved graph, so LβU0 is not zero; it carries a curvature term. The cited boundary Harnack applies to ratios of Lβ-harmonic functions vanishing on a slit. Unless the authors show how the curvature error is absorbed and the constants stay uniform under the scaling (5.10), the induction in Theorem 5.2 lacks a starting point. This is not a proven contradiction, but it is a missing verification in the written proof.

Theorem 2.6 is also an outline. The induction in Section 6 uses H/U ∈ C^α without fully justifying why the right-hand side in (6.2) preserves the boundary Harnack. Minor issues: many typos and notational slips, and the construction of Q', P' in Theorem 5.2 is compressed.

Who this is for: researchers in nonlocal free boundary problems, especially those working on fractional obstacle and Bernoulli problems. It deserves a serious referee, with the referee asked to focus on Sections 5 and 6 and to demand precise hypotheses on the imported JN17 results. If the C^{1,α}→C^{2,α} step survives scrutiny, that alone is publishable, even if the C∞ part ends up relying on [BROW25].

Recommendation: send it to referees rather than desk-reject. Ask for a standalone proof of Theorem 5.1 and a fuller proof of Theorem 2.6 before acceptance.

Referee Report

3 major / 5 minor

Summary. The paper proves C^∞ regularity of flat free boundaries for the one-phase fractional Laplacian problem (1.2) for every 0<s<1. After the Caffarelli–Silvestre extension, the problem is reformulated as (1.5). The proof has two main steps: Section 4 raises the known C^{1,α} flatness result of De Silva–Savin–Sire to C^{2,α} by paraboloid approximation, and Section 5 bootstraps C^{2,α} to C^∞ by studying the linearized problem (5.8) and showing that the quotients u_i/u_n inherit the regularity of the free boundary. The paper also states and proves an expansion theorem for the flat linearized problem (Theorem 2.6), which is used in the bootstrap.

Significance. If the proof is completed, the theorem settles the natural conjecture that locally flat free boundaries are C^∞ for the whole range 0<s<1, extending the s=1/2 result of De Silva–Savin. The Caffarelli–Silvestre strategy is a genuine alternative to the recent operator-based approach in [BROW25], and the construction of the paraboloid family V_{S,a,b}, the domain-variation estimates, and the two-step bootstrap are substantial. The paper is carefully organized and imports standard theorems in a non-circular way. However, the current version contains a load-bearing gap: the key bootstrap input Theorem 5.1 is asserted to follow from [JN17] without verifying that the curved version of U0 satisfies the hypotheses of the cited slit-domain Harnack theorems.

major comments (3)

[§5, Theorem 5.1] The assertions (5.2)–(5.4) are not established by the cited results. In (3.1), U0 is defined through the signed distance to the curved surface F(u). Even for the minimal function V=U0, the computation in Proposition 3.5 (see (3.11)) gives Lβ U0 = -|y|^β κ ∂_t U plus lower-order terms, so U0 is not Lβ-harmonic when F(u) has nonzero curvature. Theorem 7.3, imported from [JN17, Thm 1.3], requires both functions u and v to satisfy Lβu=Lβv=0 in the slit domain. Therefore Theorem 7.3 cannot be applied directly to u/U0, and the statement 'Note that (5.2) and (5.3) follow from [JN17, Proposition 3.2 and Lemma 7.2]' is not justified. The proof must either absorb the curvature terms into a controlled right-hand side of size |y|^β U0/r with constants uniform under the scaling (5.10), or provide another derivation. This is load-bearing because (5.2)–(5.4) feed into Lemma 5.4, Lemma 5.5, and Theorem
[§6, Theorem 2.6] The proof of the expansion theorem is only an outline, and one step appears to invoke a theorem whose hypotheses are not met. In Substep 1 of the induction, the function H solves div(|y|^β ∇H) = |y|^β U_t f, i.e. H is not Lβ-harmonic; nevertheless the text says 'By Theorem 7.3 Harnack inequality in slit domain, we have H/U ∈ C^α(B_{1/2})'. Theorem 7.3 applies to ratios of Lβ-harmonic functions vanishing on the slit. A version with a non-zero right-hand side, or a different argument, is needed. The final Step 2 also asserts Lβ(U0(P_{k+1}-P_k))=0 by passing to a limit, but the convergence is only uniform and the equation is degenerate; this requires justification. Since Theorem 2.6 is used in Proposition 4.2 and Lemma 5.6, the gap affects the bootstrap.
[§5, Theorem 5.2] In the proof of the main claim, after applying Lemma 5.6 and Theorem 2.6 the paper asserts the existence of polynomials ~Q and ~P with the quantitative bound |~w - ~Q - r~P| ≤ (1/4)ρ^{k+2+α} + Cρ^{k+3}. Lemma 5.6 only provides compactness of the rescaled sequence, not a quantitative rate. The argument also needs uniform solvability of the linear systems (5.15)–(5.16) for the polynomial coefficients under the scaling (5.10), with constants independent of δ. This is not proved. Because this step is the induction that upgrades the polynomial approximation from scale ρ^m to ρ^{m+1}, it is essential for the C^∞ conclusion.

minor comments (5)

[Throughout] There are numerous typos and repeated words that impede reading: 'following following' twice in the Introduction, 'viscoisity' in Definition 2.5, 'vaompare' before Lemma 3.9, 'inequlity' in Theorem 7.2, 'sophicated' in Lemma 3.10, and 'Theorem 5.12' in the proof of Lemma 3.10 should probably be 'Theorem 7.2'.
[§3, Proposition 3.5] The displayed condition before (3.11) reads 'If a+b−trM ≥ C0δ^2' but the hypothesis of the proposition and the preceding line use a/(2s)+b−trM. This is a typo in a condition that is used later.
[§5, Lemma 5.5] The claim 'osc_{B_{δ0}} w ≤ 2−δ0' should presumably be '2^{-δ0}' or a similar expression; as written the subsequent inequality (1−δ0/2)osc_{B_1}w suggests the exponent is intended.
[§5, (5.9) and (5.13)] The notation for the polynomial Q is confusing: (5.9) writes Q(x)=Q(z') but (5.13) writes Q(x')=q_μ y^μ, using y for the horizontal variable while y is also the vertical variable throughout the paper. Please use a consistent symbol for the tangential coordinates.
[§7, Theorem 7.2] The statement of the boundary Harnack inequality appears garbled: the normalization u(x0,y0)=v(x0,y0) and the conclusion ||v/u||_{C^η}≤C need clarification, since the boundary condition on Σ is written with Σ = Q_{1/2}^n(x0)×{y0+1/2}, which looks like a slice rather than the relevant boundary portion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main induction is grounded in external regularity/Harnack inputs, not in the target theorem.

full rationale

The paper's derivation does not reduce to its own inputs. The base regularity input is the external theorem of De Silva–Savin–Sire ([DSSS14, Theorem 1.1]) that flat free boundaries are C^{1,α}; the bootstrap in Section 5 imports expansion estimates from Jhaveri–Neumayer ([JN17, Proposition 3.2 and Lemma 7.2]) and Schauder-type estimates from [STV21a] and [CS16]. These are independent external results, not reformulations of Theorem 1.2. Theorem 5.1's assertion that (5.2)–(5.3) follow from [JN17] is the only potentially load-bearing external step, but an external citation whose hypotheses may or may not hold in the curved-boundary setting is a correctness risk, not a circularity: the cited results are not derived from the conclusion being proved, and the paper does not fit any parameter to force the quotient estimates. No fitted input is relabeled as a prediction, no uniqueness theorem from the author's own prior work is invoked, and no known empirical result is merely renamed. The skeptical concern that U0 is not Lβ-harmonic when F(u) is curved is a legitimate hypothesis-checking issue for the imported [JN17] estimates, but it does not make the derivation circular. The C∞ bootstrap is therefore self-contained relative to its stated external assumptions.

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

The paper is a pure regularity proof, so there are no fitted numerical parameters and no invented physical or geometric entities. The load-bearing content comes from imported analytic estimates, especially boundary Harnack and Schauder theory, which are treated as axioms in the sense that they are not rederived here.

assumptions (5)

domain assumption Caffarelli-Silvestre extension: the nonlocal problem can be replaced by the local degenerate problem Lβu=0 in one more dimension.
Used throughout the paper to pass from (1.2) to (1.5); this is standard for the fractional Laplacian but is a load-bearing structural assumption.
domain assumption DSSS14 Theorem 1.1: flat free boundaries are C^{1,α}.
The starting point of the paper; it is cited and not reproved.
domain assumption Boundary Harnack inequalities in slit domains (JN17 Theorem 1.3, quoted as Theorem 7.3).
Used in Sections 3-6 to control quotients of solutions near the free boundary slit; the quantitative dependence on boundary regularity is not checked in the text.
domain assumption Schauder and Harnack estimates for Lβ (STV21a, CSS07, FKS82; quoted as Theorems 7.1, 7.5, 7.6 and Proposition 7.4).
Used in Lemmas 4.1, 5.5, and Section 6 to obtain C^{1,t} and C^{1,1} estimates; these are imported results.
standard math Whitney extension theorem for pointwise C^{k+2,α} data (DS15 Theorem 7.1).
Used in Section 5 to extend polynomials from the free boundary to the ambient space.

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Pith. "Pith review of $C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem." pith.science (2026). https://pith.science/paper/2HVAQ6HG

@misc{pith2026250900609,
  author       = {Pith},
  title        = {Pith review of: $C^\infty$ Regularity for the free boundary of one-phase Fractional Laplacian problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HVAQ6HG}},
  note         = {Machine review of arXiv:2509.00609}
}

abstract

We consider a one-phase free boundary problem involving fractional Laplacian $(-\Delta)^s$, $0<s<1$. D. De Silva, O. Savin, and Y. Sire proved that the flat boundaries are $C^{1,\alpha}$. We raise the regularity to $C^{\infty}$, extending the result known for $(-\Delta)^{1/2}$ by D. De Silva and O. Savin.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

H. W. Alt and L. A. Caffarelli. Existence and regularity for a minimum problem with free boundary. Journal für die reine und angewandte Mathematik , 1981(325):105--144, 1981

work page 1981

[2] [2]

Obstacle problems for integro-differential operators: Higher regularity of free boundaries

Nicola Abatangelo and Xavier Ros-Oton. Obstacle problems for integro-differential operators: Higher regularity of free boundaries. Advances in Mathematics , 360:106931, 2020

work page 2020

[3] [3]

Barrios, A

B. Barrios, A. Figalli, and X. Ros-Oton. Free boundary regularity in the parabolic fractional obstacle problem. Communications on Pure and Applied Mathematics , 71:2129--2159, 2018

work page 2018

[4] [4]

Barrios, A

B. Barrios, A. Figalli, and X. Ros-Oton. Global regularity for the free boundary in the obstacle problem for the fractional laplacian. American Journal of Mathematics , 140:415--447, 2018

work page 2018

[5] [5]

Higher regularity in nonlocal free boundary problems, 2025

Begoña Barrios, Xavier Ros-Oton, and Marvin Weidner. Higher regularity in nonlocal free boundary problems, 2025

work page 2025

[6] [6]

Fully nonlinear elliptic equations , volume 43

Luis A Caffarelli and Xavier Cabr \'e . Fully nonlinear elliptic equations , volume 43. American Mathematical Soc., 1995

work page 1995

[7] [7]

Caffarelli and A

L. Caffarelli and A. Figalli. Regularity of solutions to the parabolic fractional obstacle problem. Journal für die reine und angewandte Mathematik , 680:191--233, 2013

work page 2013

[8] [8]

Caffarelli, X

L. Caffarelli, X. Ros-Oton, and J. Serra. Obstacle problems for integro-differential operators: regularity of solutions and free boundaries. Inventiones Mathematicae , 208:1155--1211, 2017

work page 2017