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On boundary regularity for the fractional p-Laplacian with unbounded reactions

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Pith's one-line read Solutions of the fractional p-Laplacian with L^q forcing stay Hölder up to the boundary, almost optimally.

desk verdict Solid, usable extension of boundary Hölder theory for the fractional p-Laplacian from bounded data to L^q reactions, almost optimal and ready to cite. read the letter →

arxiv 2607.28436 v1 pith:AHH47VZC submitted 2026-07-30 math.AP

Antonio Iannizzotto , Sunra Mosconi This is my paper · ORCID
classification math.AP MSC 35R1147H1135A15
keywords fractionalp-LaplacianboundaryregularityHöldercontinuityunboundedreactionsfinenonlocalDirichletproblemCampanatoestimates
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 Dirichlet problem for the s-fractional p-Laplacian with a reaction that is merely integrable to some power q, not necessarily bounded. It proves that the unique weak solution is Hölder continuous all the way to the boundary of a smooth domain, with an explicit exponent that depends on how large q is. When the forcing is summable enough (q larger than N/s), the solution itself is s-Hölder and the quotient of the solution by the s-power of distance to the boundary extends continuously in a Hölder fashion. These statements recover the known linear picture and are shown to be essentially sharp by explicit one-dimensional and higher-dimensional examples. The result matters because global Hölder control and fine boundary behaviour are the ingredients that unlock comparison principles, bifurcation theory, and existence arguments for nonlocal nonlinear equations with rough data.

What carries the argument

Campanato mean-oscillation estimates on (s,p)-harmonic extensions of u inside boundary balls, combined with fractional Hardy inequalities and refined monotonicity of the fractional p-Laplacian (degenerate and singular cases treated separately), which transfer interior oscillation decay and barrier controls up to the boundary.

What would settle it

Produce a C^{1,1} domain, parameters p,s,q in the stated range, and an L^q reaction whose unique solution fails to be C^α up to the boundary for some α below the claimed threshold, or whose quotient by d_Ω^s fails to be bounded when q > N/s.

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Extended reading notes

Core claim

For the homogeneous nonlocal Dirichlet problem driven by the s-fractional p-Laplacian with f in L^q, the unique solution u belongs to C^α up to the boundary for every α less than or equal to s that is strictly below p'(s − N/(p q)) when N/(p s) < q ≤ N/s, and for α = s when q > N/s; moreover, when q > N/s the quotient u/d_Ω^s admits a Hölder continuous extension to the closed domain, with a quantitative estimate in terms of the L^q-norm of f.

Load-bearing premise

The domain must have a C^{1,1} boundary so that exterior-ball conditions, distance-function regularity, and the fractional Hardy inequality all hold with uniform constants.

Editorial extensions

If this is right

  • Global C^α estimates become available for nonlocal p-Laplace equations with merely integrable right-hand sides, removing the classical L^∞ assumption.
  • When q > N/s the fine boundary regularity of u/d_Ω^s supplies a continuous fractional normal derivative on ∂Ω.
  • Existence, comparison, and bifurcation results previously limited to bounded reactions extend immediately to L^q data.
  • The almost-optimal exponents match the linear fractional Laplacian, confirming that nonlinearity does not destroy the boundary Hölder scale.

Reading between the lines

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

  • The undetermined Hölder exponent for u/d_Ω^s when q > N/s is likely improvable to the linear value s − N/q by a more precise barrier iteration.
  • The same oscillation-plus-Hardy scheme should adapt to variable-order or anisotropic fractional p-Laplacians once a corresponding Hardy inequality is available.
  • Interior C^{ᾱ} theory already known for weaker Lorentz data may combine with the present boundary argument to yield global regularity under those weaker assumptions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the Dirichlet problem for the s-fractional p-Laplacian with homogeneous exterior data and reaction f in L^q(\Omega), q > N/(ps). Theorem 1.3 asserts global Hölder continuity of the unique weak solution u up to the boundary: u \in C^\alpha(\Omegā) for every \alpha \le s with \alpha < p'(s - N/(pq)) when N/(ps) < q \le N/s, and for \alpha = s when q > N/s, with the natural estimate in terms of \| f\|_{L^q}^{1/(p-1)}. Theorem 1.5 asserts that if q > N/s then the quotient u/d_\Omega^s admits a C^\alpha extension to \Omegā for some (undetermined) \alpha \in (0,s], again with a corresponding estimate. The proofs combine Campanato oscillation estimates, (s,p)-harmonic extensions, barrier constructions, fractional Hardy inequalities, and separate monotonicity formulas in the degenerate and singular regimes, building on known interior regularity and the bounded-reaction boundary theory.

Significance. The results give a nearly complete nonlinear analogue of the linear scheme (1.4) for unbounded reactions, filling a clear gap between the interior Hölder theory (Brasco–Lindgren–Schikorra, Garain–Lindgren) and the global/fine-boundary theory available only for L^\infty data. The almost-optimality examples (1.4 and Appendix A) and the clean separation of the two regimes q \lessgtr N/s make the contribution sharp and useful for applications that rely on boundary behaviour (comparison, bifurcation, extremal solutions). The technical apparatus—especially the weighted monotonicity via Hardy and the iterative Campanato scheme that upgrades from L^\infty to any subcritical Hölder exponent—is carefully adapted and of independent interest.

minor comments (6)
  1. [Title] The title page header reads “ON BOUNDAR Y REGULARITY”; correct the spacing.
  2. [§1.1] Page 2, line after Example 1.1: “and and hence” should be “and hence”.
  3. [Theorem 1.5 / §4] In the statement of Theorem 1.5 the Hölder exponent \alpha is left completely undetermined. A brief remark on whether any explicit lower bound (even in terms of N,p,s,q only) can be extracted from the iteration in §4 would help readers who need a concrete modulus.
  4. [Lemma 2.7] Lemma 2.7 is quoted from earlier work; a one-line indication that the constant C depends only on M,\beta,\nu (and not on further geometric data of \Omega beyond C^{1,1}) would make the interpolation in the proof of Theorem 1.3 for q > N/s fully self-contained.
  5. [Appendix A] Appendix A treats only p = 2. A short sentence clarifying that the same construction is expected to work for p \neq 2 (or why it does not) would round out the optimality discussion.
  6. [§1.4] Several constants are labelled C_\Omega, C_\alpha, C_\epsilon without a uniform convention; a single sentence in §1.4 stating the dependence convention would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: global and fine-boundary claims are derived by standard nonlocal estimates, not forced by definition or self-citation loops.

full rationale

Theorems 1.3 and 1.5 are obtained from Campanato oscillation control on (s,p)-harmonic extensions, monotonicity (Lemmas 2.14–2.15), fractional Hardy (Theorem 2.12), barrier comparison (Prop. 3.1), and an approximation/interpolation argument. Prior boundary results for bounded f ([17,19]) and barriers ([18]) by overlapping authors are used as black-box inputs for the bounded case and for qualitative continuity of u/d_Ω^s; they do not encode the target statements for unbounded L^q reactions, nor do they force the Hölder exponents. Interior theory is external ([4,12]). No fitted parameters, no self-definitional identities, and no uniqueness imported to forbid alternatives. The C^{1,1}/EBC hypothesis is stated and used only where geometry is needed. The undetermined α in Theorem 1.5 and the failure to reach endpoint C^{ᾱ} are openly limitations, not circular reductions. Derivation is self-contained against the stated assumptions.

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

The paper works entirely inside standard fractional Sobolev and nonlinear potential theory. No free parameters are fitted. The load-bearing external inputs are classical embedding/regularity theorems, the exterior-ball geometry of C^{1,1} domains, and previously proved interior Hölder and barrier results. No new physical or mathematical entities are postulated.

assumptions (5)
  • standard math Interior Hölder regularity for the fractional p-Laplacian with L^q data (Brasco–Lindgren–Schikorra; Garain–Lindgren): solutions are locally C^δ for every δ<ᾱ.
    Imported as Theorem 2.4 / Corollary 2.5 and used as the starting point for boundary iteration.
  • domain assumption C^{1,1} domains satisfy the exterior ball condition and admit unique nearest-boundary-point projections in a tubular neighborhood (Lemma 2.10).
    Required for the fractional Hardy inequality, barrier construction, and distance-function regularity near ∂Ω.
  • standard math Fractional Hardy inequality on domains with finite inradius and exterior ball condition (Dyda; Brasco–Cinti).
    Theorem 2.12; used to convert Gagliardo seminorms into weighted L^p norms of u/d_Ω^s in the monotonicity arguments.
  • standard math Boundary Cs regularity and fine boundary regularity for bounded reactions (Iannizzotto–Mosconi–Squassina).
    Theorem 2.6; used as the qualitative starting point for the approximation argument that removes the L^∞ assumption on f.
  • standard math Weak comparison and nonlocal superposition principles for (−Δ)_p^s.
    Lemmas 2.1–2.2; standard and used throughout barrier and extension arguments.

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Pith. "Pith review of On boundary regularity for the fractional p-Laplacian with unbounded reactions." pith.science (2026). https://pith.science/paper/AHH47VZC

@misc{pith2026260728436,
  author       = {Pith},
  title        = {Pith review of: On boundary regularity for the fractional p-Laplacian with unbounded reactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHH47VZC}},
  note         = {Machine review of arXiv:2607.28436}
}
abstract

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega\subset\mathbb{R}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega)$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-H\"older continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/ps<q\le N/s$, and $\alpha=s$ if $q>N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a H\"older continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

Figures

Figures reproduced from arXiv: 2607.28436 by the authors.

Figure 1
Figure 1. The interior and exterior tangent balls. Lemma 2.11. Let Ω ⊂ R N be an open bounded set satisfying EBC(ρ) and with inradius RΩ < ∞. For any α > 0, there exists C > 1 depending on N, α s.t. for all x0 ∈ Ω min n 1, ρ N RN Ω o C −1 d α Ω (x0) ⩽ Z Ωc dx |x − x0|N+α ⩽ C d α Ω (x0) . Proof. Let dΩ(x0) = r > 0. Also, let x¯ ∈ ∂Ω be s.t. |x0 − x¯| = r, and Bρ(y0) be exteriorly tangent to ∂Ω at ¯x. To prove the left hand sid… view at source ↗
Figure 2
Figure 2. The set DR(¯x) in grey and the other auxiliary points. Lemma 3.3. Let x¯ ∈ ∂Ω, 0 < R < diam(Ω), u, v ∈ W s,p 0 (Ω) with being the (s, p)-harmonic extension of u in DR(x¯). Also, let 0 ⩽ α < min{1, p′ s} and Hα(u) be defined by (2.3). Then, there exists C > 0 depending on the data (but not on α), s.t. for all 0 < r ⩽ R osc Dr(¯x) v ⩽ CHα(u)R α  r R s . Proof. First we point out that (3.5) ∥u∥L∞(D2R(¯x)) ⩽ Hα(u)(2R)… view at source ↗
Figure 3
Figure 3. The set DR(x0) in gray and the other auxiliary points. Proof. Let ρ¯ be given by Lemma 2.10 and assume, without loss of generality, that 0 < ρ <¯ 1. For ε given in Proposition 3.1, define ˜ρ, depending on the data, as ρ˜ = ερ¯ 4 ∈ (0, 1). Now fix x0 ∈ Ωρ˜ and 0 < R < ρ˜, and assume u/d s Ω ∈ L∞(Ω \ DR(x0)) (otherwise there is nothing to prove). We recall that v ∈ W s,p 0 (Ω) is the unique solution of (3.1). Further,… view at source ↗

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Cited by 1 Pith paper

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