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arxiv: 2411.16296 · v4 · submitted 2024-11-25 · 🧮 math.CA · math.AP

Revisiting the Lavrentiev Phenomenon in One Dimension

Wiktor Wichrowski This is my paper

Pith reviewed 2026-05-23 17:18 UTC · model grok-4.3

classification 🧮 math.CA math.AP
keywords Lavrentiev phenomenoncalculus of variationsone dimensioncounterexampleproof inconsistencyvariational functionalinfimum equality
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The pith

Inconsistencies in Lavrentiev's original proof of the absence of the Lavrentiev phenomenon in one dimension are exposed by a counterexample, and a new concise proof establishes the theorem.

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

The paper examines the theorem that the Lavrentiev phenomenon does not appear in one-dimensional problems in the calculus of variations. It shows that the original argument from 1926 contains inconsistencies by constructing a counterexample. A new, complete reasoning is then given to confirm that the infimum of the functional over smooth functions equals the infimum over a broader class. This matters because the phenomenon concerns whether minimizers can be approximated by smoother functions without raising the value of the integral. The work also adds details in an appendix to support parts of the classical argument.

Core claim

The absence of the Lavrentiev phenomenon in one dimension holds true, with the original proof containing inconsistencies that a counterexample reveals, and a new concise and complete reasoning supplied in its place.

What carries the argument

A counterexample that exposes logical inconsistencies in the original 1926 proof, paired with a new concise reasoning that directly establishes the equality of infima for the variational functional in one dimension.

If this is right

  • The Lavrentiev phenomenon remains absent for variational problems set in one dimension.
  • The original proof from Lavrentiev contains inconsistencies that invalidate parts of its reasoning.
  • A self-contained alternative proof now exists that avoids those inconsistencies.
  • Additional details supplied in the appendix strengthen supporting arguments from the classical paper.

Where Pith is reading between the lines

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

  • The counterexample technique might be reusable on other historical proofs in the calculus of variations that rely on similar approximation steps.
  • The new reasoning could serve as a template for checking absence of the phenomenon in related one-dimensional settings with different growth conditions.
  • Clarifying the one-dimensional case may help isolate exactly which features of higher-dimensional problems allow the phenomenon to appear.

Load-bearing premise

The counterexample actually identifies a genuine logical inconsistency in the original proof rather than an inconsequential gap that leaves the theorem intact.

What would settle it

A specific one-dimensional functional for which the infimum over C^1 functions differs from the infimum over absolutely continuous functions would show the central claim is false.

Figures

Figures reproduced from arXiv: 2411.16296 by Wiktor Wichrowski.

Figure 1
Figure 1. Figure 1: In blue: Plot of y = 1 4 √ x. In red: Plot of y = 1 2 √ x. In green: Tangent line to plot of y = 1 4 √ x in point x0. properties outlined by Lavrentiev in his paper. Some of them have been reformulated for improved readability. First, Lavrentiev required that f is increasing and convex with a minimum at 0 that is greater than 1: (I) d 2 f dx2 > 0 and min x∈R f(x) = f(0) ≥ 1 [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 2
Figure 2. Figure 2: In blue: Plot of y = 1 4 √ x. In red: Plot of y = 1 2 √ x. In green: Example of y(·), with the first intersection point marked. The vertical line at p(x0) indicates cases: for functions that do not intersect with y = 1 2 √ x, we focus on the intersection point relative to the position of y = 1 4 p p(x0). We now present a new, alternative proof for this example. Example 2.2 (Lavrentiev 1926). Consider the f… view at source ↗
Figure 3
Figure 3. Figure 3: In blue, there is graph of u ′ , while on the x-axis in green the set Pε is marked. For any partition B = {B1, B2, . . . , BN } of [0, 1] into N ∈ N intervals, there exists a set Bi containing x, y ∈ Pε such that u ′ (x) = 1 and u ′ (y) = −1. That contradicts the assumption that we can expand the division of Pε into a finite number of intervals piece-wise disjoint. In this example, Pε is specifically chose… view at source ↗
read the original abstract

We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem dealing with the absence of the Lavrentiev phenomenon, a cornerstone issue in the calculus of variations. We point out some inconsistencies in the original proof by providing a counterexample and supply the result with a new, concise, and complete reasoning. In the appendix, we also provide additional details to supplement the original proof.

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 / 1 minor

Summary. The manuscript revisits Lavrentiev's theorem asserting the absence of the Lavrentiev phenomenon in one dimension. It identifies inconsistencies in the original proof by supplying a counterexample, provides a new concise and complete reasoning establishing the result, and includes an appendix with supplementary details to the original argument.

Significance. If the counterexample establishes a genuine logical inconsistency requiring a new proof and the supplied reasoning is correct, the work would clarify a foundational result in the calculus of variations. The paper supplies an independent counterexample and new reasoning rather than deriving from the original quantities.

major comments (2)
  1. [Section presenting the counterexample] The central claim requires demonstrating that the counterexample blocks a specific, non-repairable inference in Lavrentiev's original argument (rather than an auxiliary estimate or gap that leaves the main comparison of infima intact). The manuscript must explicitly identify which precise step fails and why no local repair suffices.
  2. [Section containing the new reasoning] The new concise reasoning must be shown to be complete and independent; without explicit verification that it avoids the inconsistencies exposed by the counterexample while establishing the absence of the phenomenon, the asserted need for an entirely new proof is not yet load-bearing.
minor comments (1)
  1. [Appendix] Cross-references between the new reasoning, the counterexample, and the appendix could be made more explicit to aid verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting areas where our arguments can be made more explicit. We agree that strengthening the connections between the counterexample and Lavrentiev's original steps, as well as verifying the independence of the new proof, will improve the manuscript. We will incorporate these clarifications in a revised version.

read point-by-point responses

  1. Referee: [Section presenting the counterexample] The central claim requires demonstrating that the counterexample blocks a specific, non-repairable inference in Lavrentiev's original argument (rather than an auxiliary estimate or gap that leaves the main comparison of infima intact). The manuscript must explicitly identify which precise step fails and why no local repair suffices.

    Authors: We will revise the counterexample section to explicitly name the precise inference in Lavrentiev's argument that is invalidated (the step relying on an estimate that our counterexample shows cannot hold in general). We will also add a short paragraph explaining why this inference is central to the original comparison of infima and why a local repair would require assumptions or techniques outside the scope of the 1920s argument, thereby justifying the need for an independent proof. revision: yes

  2. Referee: [Section containing the new reasoning] The new concise reasoning must be shown to be complete and independent; without explicit verification that it avoids the inconsistencies exposed by the counterexample while establishing the absence of the phenomenon, the asserted need for an entirely new proof is not yet load-bearing.

    Authors: In the revised manuscript we will insert a dedicated verification paragraph immediately after the new proof. This paragraph will (i) list the specific inconsistencies exposed by the counterexample, (ii) confirm that none of those steps are used in our argument, and (iii) outline why the new reasoning directly establishes the equality of the two infima under the stated hypotheses. This will make the independence and completeness explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; independent counterexample and new proof supplied

full rationale

The paper's central contribution consists of an explicit counterexample exposing claimed inconsistencies in Lavrentiev's 1926 argument together with an entirely new, self-contained reasoning establishing the absence of the Lavrentiev phenomenon in one dimension. Neither the counterexample construction nor the new proof reduces to a redefinition of its own inputs, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The appendix merely supplies supplementary details to the original work; the main derivation chain remains independent of the critiqued source and does not invoke any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a proof correction in pure mathematics and introduces no fitted parameters, new entities, or ad-hoc axioms beyond standard background results in the calculus of variations.

axioms (1)
  • standard math Standard theorems on lower semicontinuity and existence of minimizers in one-dimensional Sobolev spaces
    The paper relies on these background results to state the theorem being revisited.

pith-pipeline@v0.9.0 · 5576 in / 1153 out tokens · 68118 ms · 2026-05-23T17:18:38.088035+00:00 · methodology

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Reference graph

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