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arxiv: 2604.18571 · v2 · submitted 2026-04-20 · 🧮 math.AP

A strong-type unique continuation principle for the fractional p-Laplacian

Florian Grube This is my paper

Pith reviewed 2026-05-10 03:35 UTC · model grok-4.3

classification 🧮 math.AP
keywords unique continuation principlefractional p-Laplacianstrong solutionsnonlinear Schrödinger equationnonlocal operatorspartial differential equations
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The pith

The fractional p-Laplacian satisfies a strong unique continuation principle for a range of s and p.

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

The paper establishes a strong-type unique continuation principle for the fractional p-Laplacian operator (−Δ_p)^s. This asserts that any solution vanishing on a set of positive Lebesgue measure must vanish identically. The argument proceeds by a direct adaptation of techniques previously used for the weak form of the principle. The same conclusion applies to strong solutions of the fractional nonlinear Schrödinger equation. A reader would care because unique continuation controls the rigidity of solutions and rules out nontrivial localized behavior in nonlocal equations.

Core claim

For the fractional p-Laplacian (−Δ_p)^s and a range of the parameters s and p, every strong solution that vanishes on a positive-measure set is identically zero. The same strong unique continuation property holds for strong solutions of the fractional nonlinear Schrödinger equation. Both statements are obtained by a direct and simple adaptation of the arguments that establish the corresponding weak unique continuation principle.

What carries the argument

The strong unique continuation principle itself, which converts vanishing on a positive-measure set into global vanishing for strong solutions of the nonlocal equation.

Load-bearing premise

The adaptation of existing weak-unique-continuation arguments succeeds for the stated range of s and p without further restrictions on the regularity of the solutions.

What would settle it

A non-zero strong solution of the fractional p-Laplacian equation, inside the claimed range of s and p, that vanishes on some set of positive measure yet remains non-zero elsewhere.

read the original abstract

We provide a simple and direct proof of a strong-type unique continuation principle for the fractional $p$-Laplacian $(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear Schr\"odinger equation. We adapt the recent proofs of the weak UCP by Berger, Schilling and Prasad.

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 / 2 minor

Summary. The manuscript provides a simple direct proof of a strong-type unique continuation principle for the fractional p-Laplacian (−Δ_p)^s over a stated range of s and p, obtained by adapting the weak UCP arguments of Berger, Schilling and Prasad; the result is then extended to strong solutions of the fractional nonlinear Schrödinger equation.

Significance. If the adaptation is carried through correctly, the paper supplies an efficient route to strong UCP for a nonlocal nonlinear operator, which is useful for applications to fractional PDEs and nonlinear Schrödinger equations. The direct adaptation of existing weak-UCP proofs is a methodological strength when the details are fully verified.

major comments (2)
  1. [§3] §3 (main proof): the transition from the weak UCP (vanishing on open sets) to the strong UCP (vanishing on positive-measure sets) is load-bearing for the central claim. The manuscript must explicitly show, using the integral definition of (−Δ_p)^s, that if a strong solution vanishes on a set of positive Lebesgue measure then it vanishes on an open ball; without this step the adaptation from Berger–Schilling–Prasad does not automatically yield the strong form, especially since strong solutions need not be assumed continuous a priori.
  2. [§4] §4 (extension to Schrödinger equation): the precise range of s and p for which the strong UCP holds must be stated explicitly, together with the precise notion of strong solution employed, because the nonlocal character of the operator makes the passage from measure-zero vanishing to open-set vanishing sensitive to these parameters.
minor comments (2)
  1. The abstract and introduction should include the exact interval of s and p for which the result is proved, rather than the phrase “for a range of s and p.”
  2. All citations to Berger, Schilling and Prasad should appear with full bibliographic details in the reference list.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (main proof): the transition from the weak UCP (vanishing on open sets) to the strong UCP (vanishing on positive-measure sets) is load-bearing for the central claim. The manuscript must explicitly show, using the integral definition of (−Δ_p)^s, that if a strong solution vanishes on a set of positive Lebesgue measure then it vanishes on an open ball; without this step the adaptation from Berger–Schilling–Prasad does not automatically yield the strong form, especially since strong solutions need not be assumed continuous a priori.

    Authors: We agree that an explicit justification of this transition is necessary for rigor. In the revised manuscript we will insert a short lemma (or expanded paragraph) in §3 that directly invokes the integral definition of (−Δ_p)^s. The argument shows that if a strong solution vanishes on a positive-measure set E, then the nonlocal integral vanishes in a way that forces the solution to be identically zero on some open ball; this step uses only the integrability properties of strong solutions in the fractional Sobolev space and does not rely on a priori continuity. We will verify the details against the Berger–Schilling–Prasad weak-UCP framework to ensure the adaptation is complete. revision: yes

  2. Referee: [§4] §4 (extension to Schrödinger equation): the precise range of s and p for which the strong UCP holds must be stated explicitly, together with the precise notion of strong solution employed, because the nonlocal character of the operator makes the passage from measure-zero vanishing to open-set vanishing sensitive to these parameters.

    Authors: We accept this suggestion for improved clarity. In the revised version we will state the precise range of s and p at the beginning of §4 (the range already determined by the proof in §3) and give an explicit definition of strong solution (a function belonging to the appropriate fractional Sobolev space that satisfies the equation almost everywhere via the integral form of the operator). This will make the applicability of the strong UCP to the nonlinear Schrödinger equation fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: direct adaptation of external weak UCP proofs

full rationale

The paper supplies a direct proof of the strong-type UCP for (−Δ_p)^s by adapting the weak UCP arguments of Berger, Schilling and Prasad. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear; the cited works are by distinct authors and the adaptation relies on standard nonlocal integral identities and regularity properties external to the present manuscript. The extension to strong solutions of the fractional nonlinear Schrödinger equation follows from the same adapted estimates without reduction to the paper's own inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a proof in analysis relying on standard properties of fractional operators and Sobolev spaces; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

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