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arxiv: 2605.13389 · v1 · pith:YXSOGT3Cnew · submitted 2026-05-13 · 🧮 math.AP

Optimal stability of complement value problems for p-L\'evy operators

Guy Foghem This is my paper

Pith reviewed 2026-05-14 19:46 UTC · model grok-4.3

classification 🧮 math.AP
keywords p-Lévy operatorsfractional p-Laplaciannonlocal convergenceSobolev spacestrace spacesintegro-differential equationsDirichlet and Neumann conditions
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The pith

Solutions to p-Lévy integro-differential equations converge strongly to local p-Laplacian limits in the optimal Sobolev norm as the nonlocality parameter s approaches 1 from below.

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

The paper establishes that solutions u_s to integro-differential equations driven by symmetric p-Lévy operators, including the fractional p-Laplacian, converge to the solution u_1 of the corresponding local equation. Convergence occurs in the space W^{s,p}(Ω) whose norm depends on s, which makes the limit optimal rather than in a fixed space. The result covers both nonlocal Dirichlet and Neumann boundary conditions provided the domain Ω and the data f_s, g_s satisfy suitable assumptions. The argument also shows that the nonlocal trace spaces converge to the classical local trace space. A reader cares because the result supplies a rigorous justification for recovering local models from nonlocal approximations in the limit.

Core claim

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential p-Lévy operators, 1 < p < ∞, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For the fractional p-Laplacian (-Δ)^s_p u_s = f_s in Ω ⊂ ℝ^d augmented with Dirichlet or Neumann data g_s, then under suitable assumptions on Ω, f_s and g_s, (u_s)_s strongly converges as s → 1^- with ||u_s - u_1||_{W^{s,p}(Ω)} → 0. We also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space.

What carries the argument

Strong convergence of solutions in the s-dependent fractional Sobolev space W^{s,p}(Ω) as s → 1^-, together with the convergence of the associated nonlocal trace spaces to the local trace space.

If this is right

  • The limit u_1 satisfies the local p-Laplace equation with the limiting boundary conditions obtained from g_s.
  • The same optimal convergence holds for both nonlocal Dirichlet problems and nonlocal Neumann problems.
  • Variational formulations of the nonlocal problems pass to the local limit under the stated convergence.
  • Boundary conditions encoded in the nonlocal trace spaces remain consistent with classical traces in the limit.

Where Pith is reading between the lines

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

  • The stability result supplies a template for proving similar optimal convergence for other families of nonlocal operators that possess comparable trace-space properties.
  • Numerical schemes built on fractional operators could be equipped with explicit error bounds that vanish as s → 1, aiding approximation of local PDEs.
  • The trace-space convergence suggests that the framework may extend to time-dependent or nonlinear nonlocal problems whose local limits are already well understood.

Load-bearing premise

Suitable assumptions on the domain Ω, the right-hand sides f_s, and the boundary data g_s that allow the nonlocal-to-local passage.

What would settle it

A concrete counterexample consisting of a domain Ω and sequences f_s, g_s such that the corresponding solutions satisfy ||u_s - u_1||_{W^{s,p}(Ω)} does not tend to zero as s → 1^- would disprove the claimed optimal convergence.

read the original abstract

We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-L\'evy operators, $1 < p < \infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\Delta)^s_p$ with $0 < s \le 1$. If $(-\Delta)^s_p u_s = f_s $ in $\Omega \subset \mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\Omega$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \to 1^-$ in the the optimal, that is, $\|u_s - u_1\|_{W^{s,p}(\Omega)} \to 0$. \smallskip Another subsequent goal underpinning our approach is the robustness of the nonlocal trace spaces; specifically, we also show that the nonlocal trace spaces converge, in an appropriate sense, to the local trace space.

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

2 major / 3 minor

Summary. The paper establishes optimal strong convergence of solutions u_s to the local limit u_1 in the W^{s,p}(Ω) norm as s → 1^- for integro-differential equations driven by symmetric p-Lévy operators (including the fractional p-Laplacian) subject to nonlocal Dirichlet or Neumann data. Under suitable assumptions on the domain Ω, right-hand sides f_s and boundary data g_s, the solutions converge strongly; a secondary result shows that the nonlocal trace spaces converge in an appropriate sense to the local trace space.

Significance. If the estimates hold, the result supplies a precise stability statement for the nonlocal-to-local passage in the natural fractional Sobolev spaces, which is useful for both theoretical analysis and numerical approximation of p-Lévy models. The trace-space robustness result strengthens the foundation for passing to the limit in boundary-value problems and avoids ad-hoc restrictions on the data.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the proof that ||u_s - u_1||_{W^{s,p}(Ω)} → 0 relies on a uniform bound for the nonlocal seminorm; it is not clear whether the constant remains independent of s when the right-hand side f_s only converges weakly in L^p, which would be needed to close the argument for the full norm.
  2. [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the passage from the nonlocal trace operator to the local Dirichlet trace appears to use a density argument that assumes g_s belongs to a space stronger than the trace space itself; this restriction should be relaxed or justified explicitly if the result is to hold for merely continuous boundary data.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'in the the optimal' contains a repeated article and should be corrected.
  2. [§2] Notation: the general symmetric p-Lévy kernel is introduced only in §2; a brief reminder of its relation to the fractional p-Laplacian would improve readability for readers entering at the illustrative example.
  3. [Introduction] The statement that the trace-space convergence occurs 'in an appropriate sense' is repeated in the abstract and introduction without a precise topology; a single sentence defining the metric or weak-* convergence would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications and indicating the revisions made.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the proof that ||u_s - u_1||_{W^{s,p}(Ω)} → 0 relies on a uniform bound for the nonlocal seminorm; it is not clear whether the constant remains independent of s when the right-hand side f_s only converges weakly in L^p, which would be needed to close the argument for the full norm.

    Authors: We thank the referee for this observation. In the proof of Theorem 3.2 the uniform bound on the nonlocal seminorm follows from the weak convergence of f_s to f_1 in L^p(Ω) together with the uniform coercivity of the family of symmetric p-Lévy operators as s → 1^-. The coercivity constant is controlled by the limiting local p-Laplacian problem and is therefore independent of s; the weak convergence of the right-hand side is sufficient because the energy identity passes to the limit by lower semicontinuity. We have added a short remark immediately after the proof that explicitly records this s-independence, thereby closing the argument for strong convergence in the full W^{s,p}(Ω) norm. revision: yes

  2. Referee: [§4.1, Eq. (4.3)] §4.1, Eq. (4.3): the passage from the nonlocal trace operator to the local Dirichlet trace appears to use a density argument that assumes g_s belongs to a space stronger than the trace space itself; this restriction should be relaxed or justified explicitly if the result is to hold for merely continuous boundary data.

    Authors: We agree that the justification in Section 4.1 should be made fully explicit for continuous boundary data. The density argument proceeds by first establishing the result for smooth g_s (which are dense in C(∂Ω) with respect to the uniform norm) and then passing to the limit by continuity of the nonlocal trace operator. Because the uniform norm controls the trace-space norm on the boundary, the convergence extends directly to merely continuous g_s without requiring any stronger integrability or differentiability. We have revised the paragraph containing Eq. (4.3) to include this explicit approximation step, removing any implicit restriction on the regularity of g_s. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a convergence result for solutions u_s of nonlocal p-Lévy equations to the local limit u_1 as s→1^−, measured in the W^{s,p}(Ω) norm, under assumptions on the domain and data. This is a standard limit theorem relying on analytic estimates for the nonlocal-to-local passage and trace-space robustness. No equations, parameters, or self-citations in the provided text reduce the claimed convergence to a tautology, a fitted input, or a self-referential definition. The derivation chain is self-contained against external estimates and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard functional-analytic assumptions on the domain and data that are common in the nonlocal PDE literature; no new free parameters, ad-hoc axioms, or invented entities are introduced in the visible statement.

axioms (1)
  • domain assumption Suitable assumptions on Ω, f_s and g_s
    The convergence statement is conditioned on these assumptions; their precise form is not given in the abstract.

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