A note on a Pohozaev identity for the fractional Green function

Abdelrazek Dieb; Isabella Ianni

REVIEW 1 major objections 4 minor 15 references

A note on a Pohozaev identity for the fractional Green function

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that in bounded $C^{1,1}$ domains with $N>2s$, the fractional Robin function $R_s(x)$ equals $\Gamma^2(1+s)/(N-2s)$ times the boundary integral of $(G_s(x,\cdot)/\delta^s)^2\langle\cdot-x,\nu\rangle$, extending the…

desk verdict A genuinely new fractional Pohozaev identity for the Robin function, proved mostly carefully, but the one-line y→x limit in Theorem 1.1 is a real gap that needs filling. read the letter →

arxiv 2506.02806 v1 pith:4NUMJP3H submitted 2025-06-03 math.AP

Abdelrazek Dieb , Isabella Ianni This is my paper

classification math.AP MSC 35R1135A08

keywords fractionalLaplacianGreenfunctionRobinPohozaevidentitySobolevregularityboundarytrace

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 establishes a Pohozaev-type identity for the fractional Green function: in any bounded $C^{1,1}$ domain of $\mathbb{R}^N$ with $N>2s$, the fractional Robin function $R_s(x)=H_s(x,x)$ is given by a boundary integral of the squared boundary trace of the Green function, $\frac{\Gamma^2(1+s)}{N-2s}\int_{\partial\Omega}(G_s(x,\cdot)/\delta^s)^2\langle\cdot-x,\nu\rangle\,d\sigma$. This generalizes a classical identity for the standard Laplacian to every fractional order $s\in(0,1)$. The Robin function is the quantity that controls leading-order asymptotics in fractional Sobolev-critical problems, so a formula that determines it from boundary data alone could be used to extract sharp blow-up profiles for such problems. The paper also proves a two-point version involving the regular part and its gradient, valid in the whole range $s\in(0,1)$, and a counterpart local identity.

What carries the argument

The load-bearing object is the generalized fractional integration-by-parts identity (1.10) with vector field $X=\mathrm{id}-\xi$, quoted from [8]: it converts sums of weighted gradient terms into a boundary term $\Gamma(1+s)^2\int_{\partial\Omega}(u/\delta^s)(v/\delta^s)\langle X,\nu\rangle\,d\sigma$ plus a double-integral remainder. The proof feeds the Green functions through cut-off approximations $\eta_k\psi_{\mu,x}G_s(x,\cdot)$ that vanish near the boundary and near the singularity, so the identity can be applied; then Lemmas 2.1--2.9 transfer the limits $k\to\infty$, $\mu\to0^+$, $\gamma\to0^+$ to recover exactly the boundary term and the regular parts $H_s$. Sharp two-sided estimates for $G_s$ and $\nabla G_s$ from [5, 13, 2] justify the needed integrability for these limits.

What would settle it

Take $\Omega$ to be the unit ball in $\mathbb{R}^3$ and choose $s=3/4$. The fractional Green function and its regular part are known explicitly for the ball, so one can numerically evaluate both sides of (1.7) at an off-center point $x$; agreement to machine precision would support the claim, and any systematic mismatch would refute it. A cheaper check is the endpoint test: as $s\to1^-$, the right side of (1.7) should reduce to the classical identity for the Laplacian.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for bounded $C^{1,1}$ domains and $N>2s$, the fractional Robin function $R_s(x)=H_s(x,x)$ equals the boundary integral in (1.7), so the regular part of the fractional Green function is fully determined by a boundary trace. The identity is derived as the limit $y\to x$ of a two-point identity (Theorem 1.2) that expresses $(N-2s)H_s(x,y)+\langle\nabla_y H_s(x,y),y-x\rangle$ as the same kind of boundary integral. For $s>1/2$ the two-point identity holds with an arbitrary center $\xi$; for $s\le 1/2$ it holds with $\xi=x$, and taking $\xi=x$ in the former gives the full range. The proof applies a generalized fractional Pohozaev identity to regularized cutoff versions of the Green functions and passes to the limit using sharp boundary estimates; the same method yields the local identity (1.11), which the paper notes has no known counterpart even in the classical case.

Load-bearing premise

The identity is proved only for bounded $C^{1,1}$ domains and depends on sharp two-sided estimates for the fractional Green function and its gradient near the boundary; if the boundary is less regular than $C^{1,1}$, those estimates are not available and the limiting passages that produce the identity break down.

Editorial extensions

If this is right

The regular part of the fractional Green function, and hence the Robin function, is completely determined by a boundary trace of the Green function itself; no interior information is needed beyond the domain geometry.
The two-point identity (1.8) gives an explicit formula for $(N-2s)H_s(x,y)+\langle\nabla_y H_s(x,y),y-x\rangle$ in the whole range $s\in(0,1)$.
For $s>1/2$, the stronger identity (1.9) yields separate control of both gradient terms and, by subtraction at $\xi=x$ and $\xi=y$, a new boundary formula for $\langle\nabla_x H_s(y,x)+\nabla_y H_s(x,y),x-y\rangle$.
The same scheme proves the local identity (1.11), a new result even for the classical Laplacian, refining the classical asymptotics framework.
Applied to fractional Sobolev-critical boundary value problems, the formula should give sharp asymptotic characterizations of solutions, improving the known profile results.

Reading between the lines

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

The boundary-only character of (1.7) suggests that $R_s$ satisfies stronger geometric monotonicity than is currently known; for instance, in the ball one would expect $R_s(x)$ to be radial and increasing in $|x|$, which could be tested directly from the explicit Green function.
Differentiating (1.7) under the integral sign (where justified) should produce gradient formulas for $H_s(x,\cdot)$ consistent with the representation formula in the cited work, without additional regularity assumptions.
A density argument from $C^{1,1}$ domains to rougher boundaries is a natural next step: if the boundary trace $(G_s/\delta^s)^2$ remains integrable, the identity might extend to Lipschitz domains; a counterexample there would show the regularity threshold is essential.

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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 paper delivers what the title promises: a fractional version of the Brezis-Peletier Pohozaev identity for the Robin function, plus two-point identities and a local counterpart. The main identity (1.7) is new, and the two-point identity (1.8) is the real workhorse. The proof is a careful adaptation of the Djitte-Fall-Weth integration-by-parts formula using cutoffs; Lemma 2.3 is a clean new ingredient, and Remark 2.5 gives an honest account of why s>1/2 is needed for (1.9) and why (1.8) survives for all s in (0,1). I checked the algebra around (3.21); the fundamental solution computation is fine. The local-case identity (1.11) is also new, and the proof via delta mollifiers is actually more transparent than the fractional one.

The soft spot is exactly where the stress-test note lands. Theorem 1.1 is derived from (1.8) in one sentence: fix x and let y→x. That passage needs a justification. You need Gs(y,·)/δ^s → Gs(x,·)/δ^s on ∂Ω in a way that allows passage under the integral, and you need ⟨∇y Hs(x,y), y−x⟩ → 0. Neither is shown. The estimates in Remark 2.5 concern Green function regularity in the first variable, not the boundary quotient as a function of the pole. I don't think this is fatal: the boundary quotient is well behaved by standard boundary Harnack theory, and the RHS limit should follow from Hs being C^1 in each variable off the diagonal plus continuity at the diagonal. But as written it is a genuine gap in the proof of the main advertised identity.

The rest of the paper is honest about importing Chen-Song, Kulczycki, and Bogdan-Kulczycki-Nowak bounds; that is normal practice and the stated hypotheses match those sources. The authors do not oversell, and the bibliography looks appropriate. This is a solid note for the fractional Green-function community. The main theorem is probably true, the gap is probably fillable, and the two-point identities are useful beyond Theorem 1.1. I would send it to a referee and ask that the y→x limit be treated rigorously before publication.

Referee Report

1 major / 4 minor

Summary. The paper derives a fractional analogue of the Brezis–Peletier identity. Theorem 1.2 establishes, for x≠y in Ω, the two-point identity (1.8) relating the boundary integral of (G_s(x,·)/δ^s)(G_s(y,·)/δ^s) to the regular part H_s, and a variant (1.9) valid for s>1/2. Theorem 1.1 then asserts that letting y→x in (1.8) yields the Robin-function representation (1.7). The proof of Theorem 1.2 uses the generalized fractional Pohozaev identity from [8], approximates the Green functions by cut-offs ψ_{μ,x}G_s(x,·), and passes to the limit using estimates from [2,5,7,9,13]. A local s=1 counterpart is proved in Theorem 1.3. The central issue is the one-line proof of Theorem 1.1.

Significance. If the gap in the passage y→x is filled, the results are significant: the identity (1.7) generalizes the classical Brezis–Peletier formula to all s∈(0,1), determining the fractional Robin function from a boundary trace, and the two-point identities (1.8)–(1.9) are new tools for studying the gradient of the Robin function. The proof of Theorem 1.2 is careful and self-contained up to imported estimates; the split s≤1/2 vs s>1/2 is handled with explicit integrability computations (Remark 2.5). Theorem 1.3 correctly reproduces the local analogue. Nevertheless, the main theorem is currently not proven as written.

major comments (1)

[Section 3, Proof of Theorem 1.1] The proof of Theorem 1.1 consists of the single sentence 'It is enough to observe that, fixing x∈Ω in (1.8), and letting y go to x, (1.7) follows.' This limit passage is load-bearing and is not justified. In (1.8) the left-hand side contains G_s(y,·)/δ^s under a boundary integral; passing y→x requires a trace theorem for the boundary quotient G_s(z,·)/δ^s on ∂Ω and continuity of that trace in the pole z, together with a uniform integrable bound to apply dominated convergence. The estimate (2.19) is an interior two-sided bound and does not by itself control the boundary quotient as the pole approaches x, nor is any reference given for such a trace result. On the right-hand side one also needs H_s(x,y)→R_s(x) and ⟨∇_yH_s(x,y), y−x⟩→0; these are plausible but are not stated or proved. Without this step, (1.8) alone does not imply the stated formula (1.7). I recommend that the authors supply a proof (or a precise citation) of the convergence of G_s(y,·)/δ^s to G_s(x,·)/δ^s on ∂Ω as y→x, uniformly over the boundary, and verify the RHS limit.

minor comments (4)

[Lemma 2.3] Lemma 2.3 states w∈C(Ω), but the proof uses the L∞ norm of w (e.g., in the bound (2.18)). Since Ω is bounded, the assumption should be w∈C(Ω)∩L∞(Ω) or w∈C(Ω̅), and the applications indeed have bounded w.
[Lemma 2.4] In the proof of Lemma 2.4 there is a typo: 'Lt us recall' should read 'Let us recall'.
[Section 4, Theorem 1.3] The Pohozaev identity (4.1) is applied to the solutions u_{ρ,x}, u_{ρ,y} of (4.5), which are only C^1 (W^{2,p} for all p<∞) rather than C^2. The identity should be justified for such functions, for example by approximation or by noting that (4.1) extends to W^{2,p} functions. This is a standard technicality, but the manuscript currently does not address it.
[Remark 2.5] The sentence about [3, Lemma 9] is ambiguous: the paper itself proves (2.23) for all s∈(0,1) (without uniformity in x), so clarify that only the uniform-in-x versions from [3] are restricted to s>1/2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central identity is derived from an independent external Pohozaev identity via approximation and limiting arguments; the one-line y→x passage is a possible gap, not a circular reduction.

full rationale

The proof chain is not circular. Theorem 1.2 is established by applying the generalized fractional Pohozaev identity (2.37), imported from Djitte–Fall–Weth [8], to regularized functions η_k ψ_{γ,y}G_s(y,·) and η_k ψ_{μ,x}G_s(x,·), and then passing to the limits k→∞, μ→0+, γ→0+ with the help of boundary estimates from Chen–Song [5], Kulczycki [13], Bogdan–Kulczycki–Nowak [2], and approximation lemmas from Djitte–Fall–Weth [7] and Djitte–Sueur [9]. None of these cited sources shares authors with the present paper, and none of the target identities (1.7)–(1.9) is assumed among the hypotheses. Theorem 1.1 is then obtained by fixing x in (1.8) and letting y→x. That one-sentence limit passage is asserted without detailed justification, so the proof may have an analytic gap, but a missing justification is not circularity: the boundary integral representation is not inserted as an input, and no fitted parameter or normalizing convention makes the conclusion true by definition. The only reference involving a present author, the forthcoming paper [6], is mentioned only as an intended application and is not load-bearing for any proof. Under the stated C^{1,1} and N>2s assumptions, the representation (1.7) has independent mathematical content relative to the external Pohozaev identity and estimates on which the derivation rests.

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

No free parameters or invented entities. The paper builds on a chain of known results: the fractional integration-by-parts formula [8], Green function estimates [2,5,13], and regularized approximations [7,9]. The only new ingredient is Lemma 2.3 and the final identity. No fitted constant enters the derivation.

assumptions (5)

domain assumption Generalized fractional Pohozaev identity (Djitte-Fall-Weth [8, Theorem 1.3], reproduced as Lemma 2.10) holds for the regularized Green functions.
This is the fundamental integration-by-parts identity from which the proof starts; it is imported from prior work and not re-derived.
domain assumption The fractional Green function satisfies the two-sided boundary estimate (2.19) and gradient estimate (2.21) in C^{1,1} domains, from [5,13] and [2].
Used in Remark 2.5 to establish the integrability of ∇Gs and ⟨z−x,∇Gs⟩, which is essential for the limit passages in Steps 3-4 and for the range restriction s>1/2 in (1.9).
domain assumption The regularized functions ψμ,x Gs(x,·) satisfy the regularity and limit properties from [9, Lemma 2.2] and Lemmas 2.7-2.9.
These imported lemmas control the singular integrals in the limit processes as μ→0+ and k→∞; their proofs are not reproduced.
domain assumption The boundary distance function δ is C^{1,1} near ∂Ω and the Fermi map Ψ is bi-Lipschitz, allowing a change of variables near the boundary.
This standard consequence of the C^{1,1} domain assumption is used in Lemmas 2.3 and 2.4 to estimate boundary-layer integrals.
domain assumption The product rule (2.5) for the fractional Laplacian and the approximation results from [7, Corollary 6.4] and [7, Lemma 6.8] are valid.
Lemma 2.3 is new but relies on these cited estimates for the convergence of qk near the boundary.

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Pith. "Pith review of A note on a Pohozaev identity for the fractional Green function." pith.science (2026). https://pith.science/paper/4NUMJP3H

@misc{pith2026250602806,
  author       = {Pith},
  title        = {Pith review of: A note on a Pohozaev identity for the fractional Green function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NUMJP3H}},
  note         = {Machine review of arXiv:2506.02806}
}

read the original abstract

We get a Pohozaev-type identity for the fractional Green function, which extends to the fractional setting a classical result by Brezis and Peletier. Our result complements with some more recent ones obtained by Djitte and Sueur concerning a representation formula for the gradient of the fractional Robin function.

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Works this paper leans on

15 extracted references · 10 canonical work pages

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Sidy Moctar Djitte, Mouhamed Moustapha, Tobias Weth, A generalized fractional Pohozaev identity and applications, Adv. Calc. Var., 17(1):237–253, 2024, ISSN 1864-8258,1864-8266, doi:10.1515/acv-2022-0003, available from: https: //doi.org/10.1515/acv-2022-0003

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Sidy Moctar Djitte, Mouhamed Moustapha Fall, Tobias Weth, A fractional Hadamard formula and applications,Calc. Var. Partial Differential Equations , 60(6):Paper No. 231, 31, 2021, ISSN 0944-2669,1432-0835, doi:10.1007/s00526- 021-02094-3, available from: https://doi.org/10.1007/s00526-021-02094-3

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [8]

Sidy Moctar Djitte, Mouhamed Moustapha, Tobias Weth, A generalized fractional Pohozaev identity and applications, Adv. Calc. Var., 17(1):237–253, 2024, ISSN 1864-8258,1864-8266, doi:10.1515/acv-2022-0003, available from: https: //doi.org/10.1515/acv-2022-0003

work page doi:10.1515/acv-2022-0003 2024

[2] [1]

Mousomi Bhakta, Debangana Mukherjee, Sanjiban Santra, Profile of solutions for nonlocal equations with critical and supercritical nonlinearities, Commun. Contemp. Math. , 21(1):1750099, 35, 2019, ISSN 0219-1997,1793-6683, doi:10.1142/S0219199717500997, available from: https://doi.org/10.1142/S0219199717500997

work page doi:10.1142/s0219199717500997 2019

[3] [2]

Bogdan, T

K. Bogdan, T. Kulczycki, Adam Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math. , 46(2):541–556, 2002, ISSN 0019-2082,1945-6581, available from: http:// projecteuclid.org/euclid.ijm/1258136210. 15

arXiv 2002

[4] [3]

Krzysztof Bogdan, Tomasz Jakubowski, Estimates of the Green function for the fractional Laplacian perturbed by gradient, Potential Anal., 36(3):455–481, 2012, ISSN 0926-2601,1572-929X, doi:10.1007/s11118-011-9237-x, available from: https://doi.org/10.1007/s11118-011-9237-x

work page doi:10.1007/s11118-011-9237-x 2012

[5] [4]

Peletier, Asymptotics for elliptic equations involving critical growth, inPartial differential equations and the calculus of variations, Vol

Ha ¨ ım Brezis, Lambertus A. Peletier, Asymptotics for elliptic equations involving critical growth, inPartial differential equations and the calculus of variations, Vol. I , Volume 1 of Progr. Nonlinear Differential Equations Appl. , pp. 149– 192, Birkh¨ auser Boston, Boston, MA, 1989

work page 1989

[6] [5]

Ann., 312(3):465–501, 1998, ISSN 0025-5831,1432-1807, doi:10.1007/s002080050232, available from: https: //doi.org/10.1007/s002080050232

Zhen-Qing Chen, Renming Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312(3):465–501, 1998, ISSN 0025-5831,1432-1807, doi:10.1007/s002080050232, available from: https: //doi.org/10.1007/s002080050232

doi:10.1007/s002080050232 1998

[7] [6]

Francesca De Marchis, Isabella Ianni, Alberto Salda˜ na, Uniqueness of solutions to slightly subcritical fractional Lane-Emden problems in general symmetric domains, in preparation

work page

[8] [7]

Sidy Moctar Djitte, Mouhamed Moustapha Fall, Tobias Weth, A fractional Hadamard formula and applications,Calc. Var. Partial Differential Equations , 60(6):Paper No. 231, 31, 2021, ISSN 0944-2669,1432-0835, doi:10.1007/s00526- 021-02094-3, available from: https://doi.org/10.1007/s00526-021-02094-3

doi:10.1007/s00526- 2021