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arxiv: 1907.02455 · v3 · pith:AXDBLD6Inew · submitted 2019-07-04 · 🧮 math.AP

C^(2s) regularity for fully nonlinear nonlocal equations with bounded right hand side

Hern\'an Vivas This is my paper

Pith reviewed 2026-05-25 09:04 UTC · model grok-4.3

classification 🧮 math.AP
keywords fully nonlinear nonlocal equationsC^{2s} regularityconcave operatorsconvex operatorsviscosity solutionsinterior estimatestwo membranes problemfractional Laplacian
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The pith

For concave or convex fully nonlinear nonlocal elliptic operators, bounded right-hand sides imply C^{2s} interior regularity for solutions.

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

The paper proves that if I is a fully nonlinear nonlocal elliptic operator that is either concave or convex and f is bounded in the unit ball, then any solution u of Iu equals f in the ball is C^{2s} in the concentric half-ball. This sharp interior estimate generalizes the linear theory for nonlocal equations and widens earlier nonlinear results by relying only on the concavity or convexity assumption. The same conclusion supplies a basic regularity statement for the nonlocal two membranes problem. A reader would care because the result gives the expected optimal smoothness for a large family of nonlocal equations that arise when modeling long-range interactions.

Core claim

If I is a fully nonlinear nonlocal concave or convex elliptic operator and f belongs to L^infty(B_1), then Iu = f in B_1 implies u belongs to C^{2s}(B_{1/2}).

What carries the argument

The concavity or convexity of the fully nonlinear nonlocal elliptic operator I, which supplies the structural property needed to run the comparison and viscosity arguments that produce the C^{2s} bound.

If this is right

  • Solutions gain precisely two derivatives of fractional order s when the right-hand side is merely bounded.
  • The nonlocal two membranes problem obtains interior C^{2s} regularity as a direct corollary.
  • The linear nonlocal regularity theorem of Ros-Oton and Serra extends to the concave or convex nonlinear setting.
  • Earlier results on fully nonlinear nonlocal equations are improved by dropping extra structural hypotheses on the operator.

Where Pith is reading between the lines

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

  • The C^{2s} estimate can serve as the base step for a bootstrap argument that produces higher regularity once f is assumed smoother.
  • The same viscosity techniques might adapt to operators that are only approximately concave or convex in a quantitative sense.
  • Existence theories for nonlocal equations with bounded data can now invoke this interior smoothness without additional assumptions.

Load-bearing premise

The operator I must be concave or convex in addition to being fully nonlinear, nonlocal, and elliptic.

What would settle it

Construction of a fully nonlinear nonlocal elliptic operator that is neither concave nor convex, a bounded function f, and a solution u to Iu = f that fails to be C^{2s} inside the domain.

read the original abstract

We establish sharp $C^{2s}$ interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right hand side. More precisely, we show that if $I$ is a fully nonlinear nonlocal concave or convex elliptic operator and $f\in L^\infty(B_1)$ then \[ Iu=f\quad\textrm{ in }\quad B_1 \quad \Rightarrow\quad u\in C^{2s}(B_{1/2}). \] This result generalizes the linear counterpart proved by Ros-Oton and Serra and extends previous available results for fully nonlinear nonlocal operators. As an application, we get a basic regularity estimate for the nonlocal two membranes problem.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes sharp C^{2s} interior regularity estimates for solutions of fully nonlinear nonlocal equations with bounded right-hand side. Specifically, if I is a fully nonlinear nonlocal concave or convex elliptic operator and f ∈ L^∞(B_1), then Iu = f in B_1 implies u ∈ C^{2s}(B_{1/2}). The result generalizes the linear case of Ros-Oton and Serra and is applied to obtain a basic regularity estimate for the nonlocal two membranes problem.

Significance. If the central claim holds, the result meaningfully extends regularity theory from the linear to the fully nonlinear nonlocal setting by leveraging the concavity/convexity assumption to adapt comparison and envelope arguments. The explicit application to the two membranes problem demonstrates practical utility. The work credits and builds upon prior techniques without introducing free parameters or ad-hoc reductions.

minor comments (3)
  1. [§1] §1: The introduction could more explicitly contrast the new proof strategy with the linear case of Ros-Oton-Serra to highlight where concavity/convexity is used.
  2. [Preliminaries] The statement of ellipticity for I (Definition 2.1 or equivalent) should include a brief reminder of the kernel assumptions to make the main theorem self-contained.
  3. [Application section] In the two-membranes application, the reduction step to the main theorem could cite the precise corollary or proposition used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in extending the linear theory of Ros-Oton and Serra to the fully nonlinear concave/convex setting, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a regularity implication for solutions of fully nonlinear nonlocal elliptic equations under an explicit structural hypothesis (concavity or convexity of I) that is stated upfront and is independent of the conclusion. The central claim generalizes an external linear result by Ros-Oton and Serra without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. All steps rely on comparison principles and envelope arguments that are standard in the literature and do not reduce to the target regularity statement by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the structural assumption that the operator belongs to the class of fully nonlinear nonlocal concave or convex elliptic operators together with the boundedness of the right-hand side; these are domain assumptions standard in the literature but essential to the stated implication.

axioms (1)
  • domain assumption I is a fully nonlinear nonlocal concave or convex elliptic operator
    Invoked in the main implication of the abstract as the hypothesis under which the regularity holds.

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Reference graph

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