Bernoulli problem for the fractional $p$-Laplacian

Diego Marcon; Rafael Costa

REVIEW 1 major objections 3 minor 25 references

Bernoulli problem for the fractional $p$-Laplacian

T0 review · 1 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper claims optimal order-r^s growth at the free boundary for minimizers of the fractional p-Laplacian Bernoulli problem with p≥2.

desk verdict First systematic treatment of the fractional p-Laplacian Bernoulli problem, with solid proofs for existence, Hölder regularity, and the Euler-Lagrange equation in the positivity set; the optimal-growth theorem has a repairable but real rescaling gap. read the letter →

arxiv 2607.16444 v1 pith:N5333PMS submitted 2026-07-17 math.AP math.FAmath.OC

Rafael Costa , Diego Marcon This is my paper · ORCID

classification math.APmath.FAmath.OC MSC 35R3535R1135J9235B6549J40

keywords Alt-Caffarelliproblemfractionalp-LaplacianfreeboundaryoptimalgrowthHölderregularitynonlocaltailflatnesslemmaBernoulli

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 studies the one-phase Alt–Caffarelli (Bernoulli) free boundary problem for the fractional p-Laplacian, where a fractional p-energy is penalized by the measure of the positivity set. It establishes existence of minimizers, their nonnegativity and weak subsolution property, and their local Hölder continuity. The central new claim is that at any free boundary point, a nonnegative minimizer grows at most like the distance to the free boundary raised to the power s, matching the sharp order known for the linear fractional Laplacian. This would indicate that the fractional p-Laplacian Bernoulli problem belongs to the same optimal-growth universality class as its linear counterpart.

What carries the argument

The proof combines fractional p-harmonic replacements (comparing a minimizer with the solution of the homogeneous equation in a ball with the same exterior values), an energy-gap estimate that controls the L^p difference between the minimizer and its replacement by the volume penalization term, nonlocal tail estimates (including change-of-center and replacement-tail lemmas), and a Campanato-type iteration for local Hölder regularity. The optimal growth proof relies on a flatness lemma for small penalization M and a dyadic iteration that propagates simultaneously a supremum decay and a compatible tail decay across scales.

What would settle it

Directly compute sup_{B_1} ũ for the rescaled function defined in the proof of Theorem 1.4: since ũ(x) = τ u(x0+rx), the supremum over B_1 equals τ sup_{B_r(x0)} u. Inserting τ = [(sup_{B_r} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1]^{-1} gives values up to 10^s c_{n,s,p}, not ≤ 1, unless an additional unstated cancellation occurs; checking whether any hidden estimate forces the supremum below 1 would settle the base case.

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Extended reading notes

Core claim

For the functional I_M(u) = [u]^p_{W^{s,p}} + M |{u>0} ∩ Ω| with p≥2 and fixed nonnegative exterior data, minimizers exist, are nonnegative, are weak subsolutions of the homogeneous fractional p-Laplace equation, and are locally Hölder continuous. In their positivity set they solve the equation (−Δ)_p^s u = 0. The paper's main result is the optimal free boundary growth estimate: if x0 lies on the free boundary ∂{u>0}, then for small scales r and any x near x0, u(x) ≤ (c/r^s)[Tail(u;x0,r) + (⨍_{B_{2r}} u^p)^{1/p} + 1] |x−x0|^s. This is the same r^s growth rate as in the fractional Laplacian case p=2.

Load-bearing premise

The proof of the optimal growth estimate depends on a rescaling constant τ chosen so that the rescaled function satisfies sup_{B_1} ũ ≤ 1; a direct computation shows sup_{B_1} ũ = τ sup_{B_r} u can be as large as 10^s c_{n,s,p}, generally exceeding 1, so the base case of the dyadic decay iteration is unsupported.

Editorial extensions

If this is right

If the optimal growth estimate holds, minimizers have at most C^s regularity at the free boundary, matching the threshold known for the fractional Laplacian; this is likely the optimal Hölder exponent for this problem.
The subsolution property plus Hölder continuity implies the positivity set is open, and minimizers solve the homogeneous fractional p-Laplace equation where they are positive, allowing standard free-boundary methods to be applied in the positivity set.
The flatness–iteration scheme, once established, provides a template for proving optimal growth for more general nonlinear nonlocal Bernoulli problems with kernels comparable to |y|^{-n-sp}.
The energy-gap estimate gives a quantitative way to transfer regularity from fractional p-harmonic functions to minimizers, useful beyond the Bernoulli context for other free boundary problems with nonlocal nonlinear diffusion.

Reading between the lines

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

The same dyadic-flatness strategy could likely be adapted to prove a matching nondegeneracy estimate, u ≥ c r^s, a natural companion that the paper does not address; together these would pin down the exact growth rate.
The restriction p≥2 appears technical: an analogous r^s growth might be expected for 1<p<2, but the energy-gap monotonicity inequality used here may need a different form in that range.
If the optimal growth theorem is repaired, a natural next step is to study flat free boundary points and prove higher regularity (e.g., C^{1,α}) using the growth estimate as the starting point, following the linear fractional case.
The paper's reliance on the nonlocal tail suggests that the growth estimate is non-local in character; one could test whether the result remains valid with the tail term removed in the bound, which would indicate a stronger local smoothing effect.

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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

First thing to know: this is the first systematic treatment of the one-phase Bernoulli problem for the fractional p-Laplacian, and the first three theorems are in good shape. The headline optimal-growth theorem is plausible but its proof has a real normalization gap that will need fixing.

What is actually new: existence via the direct method, nonnegativity, the weak subsolution property, local Hölder continuity by harmonic replacement plus Campanato iteration, and the fact that minimizers solve the fractional p-Laplace equation in their positivity set. These results genuinely extend the linear fractional case and the local p-case, and the citation list reflects the relevant prior work. The proofs of Theorems 1.1–1.3 look coherent to me; the tail estimates and the energy-gap lemma are standard and are used correctly.

The soft spot is Theorem 1.4. The stress-test concern is correct: the rescaling constant τ = [ (sup_{B_r} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1 ]^{-1} does not imply sup_{B_1} ũ ≤ 1. The computation gives sup_{B_1} ũ ≤ 10^s c_{n,s,p}, not 1, and the average bound used in the base case fails for the same reason. The dyadic iteration never gets started as written. This is a genuine gap in a central estimate. It is repairable: taking τ = (sup_{B_r} u + Tail(u;x0,r/2) + 1)^{-1} restores normalization and still gives the tail bound. But the paper needs to fix this before the theorem can be accepted.

Two lesser points. The comparison principle used in Lemma 5.1 is invoked without a citation; it is standard for the fractional p-Laplacian, but should be stated. The step labeled (5.5) looks like it skips the fact that v(0) ≤ sup_{B_{ι/2}} v; that is immediate, so I do not count it as a real flaw. I also did not find circularities or fitted parameters; the proofs use cited external estimates and the paper's own earlier theorems legitimately.

Bottom line: the first half of the paper is solid and worth citing, and the r^s growth rate is very likely correct. This deserves a serious referee even though my own verdict on the current version would be conditional. I would send it to review, and I hope the authors fix the rescaling rather than bury the claim.

Referee Report

1 major / 3 minor

Summary. This paper studies the one-phase Alt--Caffarelli (Bernoulli) problem for the fractional $p$-Laplacian, $p\ge 2$, with a prescribed nonnegative exterior datum and a penalization proportional to the measure of the positivity set. The authors claim existence of minimizers, their nonnegativity and subsolution property (Theorem 1.1), local H\"older continuity of minimizers (Theorem 1.2), the Euler--Lagrange equation in the positivity set (Corollary 1.3), and an optimal $r^s$ free-boundary growth estimate (Theorem 1.4). The proofs combine fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, a Campanato-type iteration, and a flatness lemma.

Significance. If all results hold, this paper would provide the first systematic free-boundary regularity theory for the Bernoulli problem driven by the fractional $p$-Laplacian, extending the known linear case $p=2$ to the full quasilinear nonlocal range. The proofs of Theorems 1.1--1.3 appear coherent and rely on cited estimates; the paper contains no fitted parameters and produces explicit quantitative bounds. The optimal growth theorem is the headline result and would be a significant contribution. However, as detailed below, the proof of Theorem 1.4 contains a normalization error in the central iterative argument. The gap is localized and appears repairable, so the manuscript is promising, but the main theorem is not established as written.

major comments (1)

[Section 5, proof of Theorem 1.4] In the proof of Theorem 1.4, the rescaling constant is defined by τ = ((sup_{B_r(x0)} u + Tail(u;x0,r/2))/(10^s c_{n,s,p}) + 1)^{-1}. The paper asserts that sup_{B_1} ũ ≤ 1 and hence (f_{B_1} ũ^p)^{1/p} ≤ 1. A direct computation gives sup_{B_1} ũ = τ sup_{B_r} u = A/(A/(10^s c)+1), where A = sup_{B_r} u; this quantity is < 10^s c, not ≤ 1. Thus the base case (5.9) of the dyadic decay is unsupported, and Lemma 5.1 cannot be applied at the first step. Since (5.9)--(5.10) are the iteration that yields the growth estimate, Theorem 1.4 is not proved as written. The defect appears repairable: for example, a normalization with τ = (A + Tail(u;x0,r/2) + 1)^{-1} makes sup_{B_1} ũ ≤ 1 and Tail(ũ;0,1/2) ≤ 1, but then the constants in Lemma 5.1 and the choice of ε must be rebalanced so that the induction step still produces sup_{B_{1/10}} ũ_k ≤ 10^{-s}. I regard this as a genuine but fixable gap, no

minor comments (3)

[Definition 2.6] Typo: 'Frational p-harmonic replacement' should read 'Fractional p-harmonic replacement'.
[Section 3, proof of Corollary 3.3] The text refers to 'Theorem 3.2'; the actual reference should be Lemma 3.2 (or Theorem 1.1). Similar cross-reference errors appear elsewhere: Section 4 uses 'Theorem 2.3' for Corollary 2.3 and 'Theorem 4.2' for Lemma 4.2; Section 5 uses 'Theorem 4.1' for Lemma 4.1; and the Euler--Lagrange result is called Corollary 1.3 in the introduction but Theorem 1.3 in Section 4.
[Section 5, equation (5.1)] The choice of c_{n,s,p} satisfies a lower-bound inequality only if 1 - 10^{-s/(p-1)} > 0; this is true for s>0, but it may help the reader to state explicitly that c_{n,s,p} is chosen sufficiently large.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation uses external estimates and internally proved theorems; the τ-normalization gap in Theorem 1.4 is a correctness issue, not a circular reduction.

full rationale

The paper's derivation chain is self-contained in the relevant sense: Theorem 1.1 uses the direct method, Theorem 1.2 is proved in Section 4 using fractional p-harmonic replacements and external estimates (Cozzi's Poincaré inequality, Di Castro–Kuusi–Palatucci regularity, fractional Sobolev embedding), and Theorem 1.4 uses Lemma 5.1 together with the previously proved Theorem 1.2. There are no fitted parameters that are subsequently renamed as predictions, and the authors do not rely on load-bearing self-citations: they cite no prior work of their own, and the strategy inspired by [23] is an external methodological reference, not an unverified premise. The normalization constant τ in the proof of Theorem 1.4 is defined in terms of sup_{B_r} u and Tail(u;x0,r/2), but the final estimate is not a tautology: the pointwise r^s decay is produced by the dyadic iteration in Lemma 5.1, and the bound on τ^{-1} is obtained from the subsolution estimate Theorem 2.1, not from the definition of τ. The manuscript does contain a serious non-circular gap, namely that the claimed normalization sup_{B_1} ũ ≤ 1 does not follow from the stated choice of τ; a direct computation gives sup_{B_1} ũ = τ sup_{B_r} u, which can be much larger than 1. This makes the base case (5.9) of the iteration unsupported, so Theorem 1.4 is not established as written. This is a mathematical correctness problem, not a circularity problem, because the claimed result is not equivalent to its inputs by construction. The gap appears repairable by adjusting τ, but the paper's headline proof is incomplete as written.

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

No empirical fitting, no invented entities. The proof rests on cited regularity theory for fractional p-harmonic functions and standard fractional Sobolev tools. The only ad hoc premise is the normalization in Theorem 1.4, which is not justified as written.

assumptions (5)

standard math Regularity estimates for fractional p-harmonic functions (local boundedness, Hölder continuity, Harnack inequality, tail estimates) as in Di Castro-Kuusi-Palatucci [15,16].
Used throughout Sections 2-5; these are external theorems, not proved in this paper.
standard math Fractional Poincaré-Friedrichs inequality and compact embedding theorems for fractional Sobolev spaces [6,17].
Used in the existence proof (Proposition 3.1) and in energy-gap estimates (Corollary 4.3, Lemma 5.1).
domain assumption Comparison principle for weak sub/supersolutions of the fractional p-Laplacian.
Invoked in Lemma 5.1 to assert u ≤ v for the p-harmonic replacement v; no reference is given in the paper.
standard math Campanato characterization of Hölder spaces.
Used in the proof of Theorem 1.2 to convert mean-oscillation decay into Hölder continuity.
ad hoc to paper The rescaling τ in Theorem 1.4 makes sup_{B_1} ũ ≤ 1.
This is a load-bearing premise of the iteration in Theorem 1.4; as written it is false, since τ sup can be of order 10^s c_{n,s,p}.

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Cite this review

Pith. "Pith review of Bernoulli problem for the fractional $p$-Laplacian." pith.science (2026). https://pith.science/paper/N5333PMS

@misc{pith2026260716444,
  author       = {Pith},
  title        = {Pith review of: Bernoulli problem for the fractional $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N5333PMS}},
  note         = {Machine review of arXiv:2607.16444}
}

abstract

We study regularity properties of minimizers for the one-phase Alt--Caffarelli problem associated with the fractional $p$-Laplacian in the range $p\geq2$. We consider minimizers of the fractional $p$-energy penalized by the measure of the positivity set, with prescribed nonnegative exterior datum. We prove existence of minimizers and derive their basic variational properties: minimizers are nonnegative and are weak subsolutions of the homogeneous fractional $p$-Laplace equation. The main regularity argument combines fractional $p$-harmonic replacements, energy-gap estimates, nonlocal tail bounds, and a Campanato-type iteration. This yields local H\"older continuity of minimizers and implies that they solve the homogeneous equation in their positivity set. Finally, we prove the optimal free boundary growth estimate, showing that the sharp order known in the linear fractional Bernoulli problem persists in the fractional $p$-Laplacian setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 1, 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. J. Reine Angew. Math., 325:105–144, 1981

1981

[2] [2]

Caffarelli, and Avner Friedman

Hans Wilhelm Alt, Luis A. Caffarelli, and Avner Friedman. Variational problems with two phases and their free boundaries. Trans. Amer. Math. Soc., 282(2):431–461, 1984

1984

[3] [3]

A geometric approach to free boundary problems, volume 68 of Graduate Studies in Mathematics

Luis Caffarelli and Sandro Salsa. A geometric approach to free boundary problems, volume 68 of Graduate Studies in Mathematics. American Mathematical Society, Prov- idence, RI, 2005

2005

[4] [4]

An extension problem related to the fractional Lapla- cian

Luis Caffarelli and Luis Silvestre. An extension problem related to the fractional Lapla- cian. Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007

2007

[5] [5]

Caffarelli, Jean-Michel Roquejoffre, and Yannick Sire

Luis A. Caffarelli, Jean-Michel Roquejoffre, and Yannick Sire. Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. (JEMS), 12(5):1151– 1179, 2010

2010

[6] [6]

Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes

Matteo Cozzi. Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes. J. Funct. Anal., 272(11):4762–4837, 2017

2017

[7] [7]

Jo˜ ao V ´ ıtor da Silva and Julio D. Rossi. The limit asp→ ∞in free boundary problems with fractionalp-Laplacians. Trans. Amer. Math. Soc., 371(4):2739–2769, 2019

2019

[8] [8]

A minimum problem with free boundary for a degenerate quasilinear operator

Donatella Danielli and Arshak Petrosyan. A minimum problem with free boundary for a degenerate quasilinear operator. Calc. Var. Partial Differential Equations, 23(1):97–124, 2005. 22

2005