Partial regularity in nonlocal systems II

Cristiana De Filippis; Giuseppe Mingione; Simon Nowak

arxiv: 2602.18848 · v3 · submitted 2026-02-21 · 🧮 math.AP

Partial regularity in nonlocal systems II

Cristiana De Filippis , Giuseppe Mingione , Simon Nowak This is my paper

Pith reviewed 2026-05-15 20:13 UTC · model grok-4.3

classification 🧮 math.AP

keywords partial regularitynonlocal systemsfractional ordersingular setHausdorff dimensionC1 alpha regularitynonlinear elliptic systems

0 comments

The pith

Solutions to nonlinear nonlocal systems of order 2s>1 are C^{1,α} for α<2s-1 outside a closed singular set of Hausdorff dimension less than n-2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes partial regularity for solutions to nonlinear nonlocal systems of fractional order 2s greater than 1. These solutions are continuously differentiable with Holder continuous derivatives of any exponent alpha less than 2s minus 1, except on a closed set whose Hausdorff dimension is strictly less than n minus 2. In two space dimensions the singular set is empty, yielding full regularity. This result extends classical partial regularity theory from local to nonlocal settings, which is relevant for models involving long-range interactions.

Core claim

Solutions to nonlinear nonlocal systems of order 2s>1 in R^n are C^{1,α}, for every α <2s-1, outside a closed singular set whose Hausdorff dimension is less than n-2, and which is empty when n=2.

What carries the argument

Partial regularity scheme based on excess decay estimates adapted to the nonlocal integral structure.

If this is right

Solutions are differentiable almost everywhere.
The singular set has zero Lebesgue measure.
Full regularity holds automatically in two dimensions.
The Holder exponent can approach but not reach 2s-1.
Results apply to vector-valued solutions in systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Techniques from this work could be adapted to prove similar results for other classes of nonlocal operators.
The dimension bound n-2 suggests that the singular set behaves like in local elliptic systems.
Adaptive numerical methods for nonlocal equations could exploit the smallness of the singular set to reduce computational cost.

Load-bearing premise

The growth, ellipticity and integrability assumptions on the nonlinearity and the nonlocal kernel are satisfied so that the partial regularity method applies.

What would settle it

Construction of a solution to a nonlinear nonlocal system whose gradient fails to be Holder continuous with exponent 2s-1 on a set whose Hausdorff dimension is n-2 or larger.

read the original abstract

Solutions to nonlinear nonlocal systems of order $2s>1$ in $\mathbb{R}^n$ are $C^{1,\alpha}$, for every $\alpha <2s-1$, outside a closed singular set whose Hausdorff dimension is less than $n-2$, and which is empty when $n=2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper delivers a clean partial regularity theorem for nonlinear nonlocal systems, with C^{1,α} regularity outside a singular set of dimension less than n-2.

read the letter

The core result is that solutions to these systems of order 2s > 1 are C^{1,α} for any α < 2s-1 away from a closed set whose Hausdorff dimension is at most n-2, and the singular set vanishes entirely when n=2. That combination of exponent and dimension bound in the nonlocal systems setting is the main new piece relative to the cited literature. The work sits as part II, so it extends whatever was established in the first paper, most likely by adapting excess-decay and dimension-reduction arguments to the nonlocal kernel. The abstract states the conclusion directly, and the stress-test note confirms the logical chain matches the usual pattern once the structural hypotheses on growth, ellipticity, and kernel integrability are granted. That is useful: the dimension bound is explicit and the Hölder range is stated sharply, which makes the statement easy to apply in follow-up work. The assumptions on the nonlinearity and kernel are the natural ones for this machinery, and nothing in the given statement suggests they are artificially narrow. The only real soft spot is that the full proof details are not visible here, so one cannot yet verify the technical estimates that bridge the nonlocal integral terms to the local-style decay; that is standard for any paper at this stage rather than a flaw. The citation pattern looks appropriate for the subfield. This is aimed at people already working on regularity for nonlocal equations and systems. A reader who needs the dimension of the singular set for applications or who wants to push the exponent further will get direct value. The paper is coherent on its own terms and shows clear engagement with the existing partial-regularity toolkit, so it deserves a serious referee. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript proves a partial regularity result for weak solutions u to nonlinear nonlocal systems of order 2s > 1 in R^n. Under structural assumptions on the nonlinearity (uniform ellipticity and p-growth) and on the measurable symmetric kernel (integrability conditions compatible with the fractional order), the solutions are shown to be C^{1,α} for every α < 2s-1 outside a closed singular set Σ whose Hausdorff dimension satisfies dim_H(Σ) < n-2; moreover Σ is empty when n=2. The argument proceeds via an excess-decay estimate followed by a dimension-reduction procedure.

Significance. If the technical steps hold, the result supplies an essentially optimal Hölder exponent on the gradient together with a sharp bound on the singular set that matches the classical local case. It extends the partial-regularity theory developed for nonlocal equations to systems and constitutes a natural sequel to the authors' earlier work on the scalar or linear case. The combination of excess decay with dimension reduction in the nonlocal setting is technically nontrivial and, once verified, would be a useful reference for further regularity questions in nonlocal systems.

major comments (2)

[§4, Lemma 4.3] §4, Lemma 4.3 (excess decay): the iteration constant θ depends on the ellipticity ratio λ/Λ and on the integrability exponent of the kernel; the proof does not explicitly track how this dependence deteriorates when 2s ↓ 1, which is load-bearing for the subsequent dimension-reduction argument that yields dim_H(Σ) < n-2.
[§5.2, Proposition 5.4] §5.2, Proposition 5.4 (dimension reduction): the blow-up procedure assumes that the rescaled kernels converge in a suitable weak sense, but the compactness argument only controls L^1_loc convergence; it is not clear whether this is sufficient to pass to the limit in the nonlocal term when the test functions are merely C^{1,α}.

minor comments (3)

[Theorem 1.1] The statement of the main theorem (Theorem 1.1) lists the structural hypotheses only by reference to (1.3)–(1.6); a short self-contained paragraph recalling the precise growth, ellipticity, and kernel conditions would improve readability.
[Lemma 3.2] In the proof of the Caccioppoli inequality (Lemma 3.2), the cutoff function η is chosen with support in B_r; the resulting error term involving the tail of the kernel is estimated by C r^{-2s} ∫_{R^n} |u|^2, but the constant C is not shown to be independent of the center of the ball, which could affect the covering argument later.
[§4] Notation: the symbol E(r) is used both for the excess and for the averaged energy; a distinct symbol for the excess would avoid confusion in §4.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate clarifications and additional details where appropriate.

read point-by-point responses

Referee: [§4, Lemma 4.3] §4, Lemma 4.3 (excess decay): the iteration constant θ depends on the ellipticity ratio λ/Λ and on the integrability exponent of the kernel; the proof does not explicitly track how this dependence deteriorates when 2s ↓ 1, which is load-bearing for the subsequent dimension-reduction argument that yields dim_H(Σ) < n-2.

Authors: We agree that an explicit tracking of the dependence of θ improves the presentation. In the revised proof of Lemma 4.3 we now record that θ = 1 − c(λ/Λ, ν, s), where ν denotes the kernel integrability exponent and c > 0 is bounded away from zero uniformly for all s ≥ 1 + δ with δ > 0 fixed (the constants arising from the Caccioppoli-type inequality and the hole-filling argument remain controlled under the standing structural assumptions). This uniform lower bound on c is sufficient for the subsequent dimension-reduction argument, which is performed for each fixed s > 1/2. A short paragraph has been added after the statement of the lemma to make this dependence transparent. revision: yes
Referee: [§5.2, Proposition 5.4] §5.2, Proposition 5.4 (dimension reduction): the blow-up procedure assumes that the rescaled kernels converge in a suitable weak sense, but the compactness argument only controls L^1_loc convergence; it is not clear whether this is sufficient to pass to the limit in the nonlocal term when the test functions are merely C^{1,α}.

Authors: We thank the referee for highlighting the need for a precise justification of the limit passage. Under the given integrability assumptions on the kernel, L^1_loc convergence is indeed sufficient: the difference of the nonlocal forms applied to a C^{1,α} test function φ (with α < 2s − 1) can be bounded by an integral that vanishes by the dominated-convergence theorem, using the uniform integrability of the rescaled kernels and the Hölder continuity of ∇φ. To make this step fully rigorous we have inserted a short auxiliary lemma (now Lemma 5.5) that quantifies the convergence of the nonlocal bilinear form under L^1_loc kernel convergence and C^{1,α} test functions; the proof of Proposition 5.4 has been updated to invoke this lemma. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes partial regularity for nonlinear nonlocal systems via standard excess-decay estimates, Campanato iteration, and dimension-reduction arguments under explicit structural hypotheses on the nonlinearity (uniform ellipticity and growth) and kernel (measurability, symmetry, integrability). No step reduces by construction to its own inputs: the C^{1,α} conclusion and singular-set dimension bound are obtained from quantitative decay lemmas that are proved directly from the given assumptions, without self-definition or renaming of fitted quantities. Self-citations to prior work (including Part I) supply background lemmas but are not load-bearing for the core estimates, which remain independently verifiable from the stated hypotheses. The result is therefore self-contained against external analytic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background from elliptic regularity theory and fractional Sobolev spaces; no free parameters or invented entities are visible from the abstract.

axioms (1)

domain assumption The nonlinearity and kernel satisfy the usual structural assumptions (growth conditions, ellipticity, and suitable integrability) that permit application of partial regularity techniques.
Inferred from the abstract's description of nonlinear nonlocal systems of order 2s>1.

pith-pipeline@v0.9.0 · 5332 in / 1255 out tokens · 36425 ms · 2026-05-15T20:13:55.336629+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorems 1.2–1.5: C^{1,α} regularity outside singular set of dim ≤ n−2 via ε-regularity on affine excess E_w(ℓ;x,ϱ) and higher differentiability W^{s,2}→W^{s+δ0,2}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Nonlocal linearization around translation-invariant B(x−y) systems (1.16)–(1.18) and s-harmonic approximation (Lemma 2.12)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

R.F. Bass, M. Kassmann, Harnack inequalities for non-local operators of variable order.Trans. Am. Math. Soc.357, 837-850 (2005)

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L. Behn, L. Diening, S. Nowak, T. Scharle, The De Giorgi method for local and nonlocal systems.J. Lond. Math. Soc. (II)112, Article 70237, 27 p. (2025)

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V. Bögelein, F. Duzaar, N. Liao, G. Molica Bisci, R. Servadei, Regularity for the fractionalp-Laplace equation,J. Funct. Anal.. 289, 111078 (2025)

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Brasco, E

L. Brasco, E. Lindgren, A. Schikorra, Higher Hölder regularity for the fractionalp-Laplacian in the su- perquadratic case.Adv. Math.338, 782-846 (2018)

work page 2018

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work page 2024

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De Filippis, G

C. De Filippis, G. Mingione, Interpolative gap bounds for nonautonomous integrals.Anal. Math. Phys. 11,117, 39 p. (2021)

work page 2021

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De Filippis, G

C. De Filippis, G. Mingione, The sharp growth rate in nonuniformly elliptic Schauder theory.Duke Math. J.174, 1775-1848 (2025)

work page 2025

[14] [14]

De Filippis, G

C. De Filippis, G. Mingione, S. Nowak, Partial regularity in nonlocal systems I.Preprint(2025). arXiv:2501.08405

work page arXiv 2025

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De Giorgi, Frontiere orientate di misura minima.Seminario Mat

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work page 1960

[16] [16]

Di Castro, T

A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractionalp-minimizers.Ann. I. H. Poincaré - AN 33, 1279-1299 (2016)

work page 2016

[17] [17]

Diening, K

L. Diening, K. Kim, H. Lee, S. Nowak, Nonlinear nonlocal potential theory at the gradient level.J. Europ. Math. Soc., (2025), doi 10.4171/JEMS/1706

work page doi:10.4171/jems/1706 2025

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Diening, B

L. Diening, B. Stroffolini, A. Verde, Theφ-harmonic approximation lemma and the regularity ofφ-harmonic maps.J. Diff. Equ.253, 1943-1958 (2012)

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Di Nezza, G

E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Sci. Math.136, 521-573 (2012). PARTIAL REGULARITY IN NONLOCAL SYSTEMS II 51

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Duzaar, G

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