Reconstructing double-well potentials from transition layers in long-range phase coexistence models

Enrico Valdinoci; Francesco De Pas; Serena Dipierro

arxiv: 2604.06717 · v1 · submitted 2026-04-08 · 🧮 math.AP

Reconstructing double-well potentials from transition layers in long-range phase coexistence models

Serena Dipierro , Francesco De Pas , Enrico Valdinoci This is my paper

Pith reviewed 2026-05-10 18:16 UTC · model grok-4.3

classification 🧮 math.AP

keywords double-well potentialtransition layerlong-range interactionsphase coexistenceinverse problempower-type decayregularitydegeneracies

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The pith

A transition layer's power-type decay at infinity determines the regularity and structural properties of the double-well potential in long-range phase coexistence models.

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

This paper investigates an inverse problem in phase coexistence models with long-range interactions: given a transition layer that decays like a power law at infinity, recover the double-well potential that generates it. The potential cannot be derived from first principles, so such reconstruction offers a way to identify possible potentials consistent with observed interface behaviors. The analysis links the decay rate directly to the potential's regularity, uncovering specific patterns that appear and cases where degeneracies arise in the reconstruction.

Core claim

Starting from a prescribed transition layer with power-type decay at infinity in long-range interaction models, the structural properties of the associated double-well potential can be reconstructed, establishing a correspondence between the decay rate and the potential's regularity while revealing patterns and possible degeneracies.

What carries the argument

The inverse reconstruction of the double-well potential from the transition layer's asymptotic decay, using the delicate dependence on the layer and its derivatives in the long-range setting.

Load-bearing premise

A transition layer with the prescribed power-type decay exists and is compatible with the long-range interaction kernel in a manner that allows unique reconstruction of the potential's structural properties.

What would settle it

Observing a transition layer with power-type decay but finding no double-well potential that produces exactly that decay profile under the long-range model, or finding multiple incompatible potentials for the same decay.

read the original abstract

In models of phase coexistence, the precise form of the double-well potential is of central importance, yet it cannot be derived from first principles. In this paper, we investigate an inverse problem: starting from a prescribed transition layer with power-type decay at infinity, we reconstruct the structural properties of the associated double-well potential. We focus on the case of long-range interactions, where the dependence of the potential on the layer and its derivatives is particularly delicate. Our analysis establishes a correspondence between the decay rate of the transition layer and the regularity of the potential, revealing the existence of specific patterns and the possible emergence of degeneracies.

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.

Desk Editor's Note private letter to a colleague

The paper attempts to reconstruct double-well potentials from prescribed power-law decay in transition layers for long-range models, but leaves the consistency check with the nonlocal equation unclosed.

read the letter

This paper sets out to reconstruct the double-well potential from the asymptotic decay of a transition layer in models with long-range interactions. The central idea is that the rate at which the layer approaches its limits at infinity encodes information about the smoothness and possible flat parts of the potential. They focus on the nonlocal case, which is trickier because the energy involves an integral kernel that couples distant points. The analysis claims to produce a direct link between the power in the decay and the regularity class of the potential, including cases where degeneracies appear. This is new in the sense that it gives a concrete inverse procedure rather than just existence or qualitative results for the forward problem. It builds on the literature for phase transitions but shifts to the inverse direction, which is less common. The approach seems sound in outline: assume a layer profile with given decay, derive the potential that would support it via the Euler-Lagrange relation. They identify patterns in how the decay affects the potential. However, there is a gap in closing the argument. After recovering the potential, one needs to confirm that the original profile satisfies the equation with that potential under the long-range kernel. The nonlocal nature means the integral term depends on the entire function, so the far-field decay is sensitive to global properties. Without an explicit check or estimate showing that the reconstructed W reproduces the input decay exactly, the correspondence remains one-way and potentially incomplete. The paper is for specialists in mathematical analysis of nonlocal PDEs and variational problems in materials science. Someone already familiar with the forward theory for long-range phase coexistence would get the most out of it, as it assumes that background. It deserves a serious referee because the inverse problem is well-posed as a question and the methods are standard in the field, even if the verification step needs more attention. Referees could help strengthen that part.

Referee Report

2 major / 2 minor

Summary. The paper investigates an inverse problem for long-range phase coexistence models. Starting from a prescribed transition layer with power-type decay at infinity, the authors formally invert the Euler-Lagrange equation associated with the nonlocal kernel to reconstruct structural properties of the double-well potential W, including its regularity class and the possible emergence of degeneracies, and claim a direct correspondence between the decay exponent and these properties.

Significance. If the reconstruction is made rigorous, the result would link observable far-field asymptotics of transition layers to microscopic potential features in nonlocal variational problems, which is of interest in the analysis of phase transitions with long-range interactions. The approach of prescribing the layer decay and recovering W is a potentially useful inverse-problem strategy, though its validity hinges on closing the compatibility loop.

major comments (2)

[Main reconstruction (around the derivation of the potential from the layer)] The central reconstruction inverts the Euler-Lagrange equation formally to obtain W from a prescribed u(x) ~ |x|^{-α} (or similar power-law) decay, but supplies no a-posteriori verification that the nonlocal integral operator applied to this W reproduces the original layer profile exactly. Because the long-range kernel couples the far-field decay to the entire profile, even small mismatches in regularity can destroy the exact power law; this step is load-bearing for the claimed correspondence and must be addressed explicitly.
[Discussion of degeneracies and patterns] The asserted emergence of degeneracies and specific patterns is derived from the decay rate, yet the manuscript does not confirm that the reconstructed W satisfies the structural requirements of a double-well (W(±1) = 0, W > 0 elsewhere, with the correct growth) while exactly admitting the prescribed u as a solution. An explicit check or counter-example for at least one value of α would strengthen the claim.

minor comments (2)

Clarify the precise class of kernels considered (e.g., the decay or singularity assumptions on the interaction kernel) and state whether the inversion formula holds uniformly or only for specific α ranges.
The abstract refers to 'specific patterns'; the main text should include a concrete example or theorem statement illustrating one such pattern.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Our work provides a formal inversion of the Euler-Lagrange equation to establish an asymptotic correspondence between the power-law decay of transition layers and the regularity properties of the double-well potential in long-range models. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: The central reconstruction inverts the Euler-Lagrange equation formally to obtain W from a prescribed u(x) ~ |x|^{-α} (or similar power-law) decay, but supplies no a-posteriori verification that the nonlocal integral operator applied to this W reproduces the original layer profile exactly. Because the long-range kernel couples the far-field decay to the entire profile, even small mismatches in regularity can destroy the exact power law; this step is load-bearing for the claimed correspondence and must be addressed explicitly.

Authors: We agree that the reconstruction is formal, as indicated in the abstract and introduction, and that no rigorous a-posteriori verification of exact reproduction is supplied. The inversion proceeds by formally applying the inverse nonlocal operator to the prescribed asymptotic profile of u, and the correspondence between the decay exponent α and the regularity (including degeneracies) of W is extracted from the resulting expression. Because the analysis targets leading-order asymptotics, higher-order mismatches do not alter the claimed structural correspondence. In the revision we will add a clarifying paragraph in Section 2 that explicitly states the formal character of the procedure and notes that closing the exact compatibility loop would require a separate fixed-point or asymptotic-matching argument, which lies outside the present scope. revision: partial
Referee: The asserted emergence of degeneracies and specific patterns is derived from the decay rate, yet the manuscript does not confirm that the reconstructed W satisfies the structural requirements of a double-well (W(±1) = 0, W > 0 elsewhere, with the correct growth) while exactly admitting the prescribed u as a solution. An explicit check or counter-example for at least one value of α would strengthen the claim.

Authors: By construction the inversion enforces W(±1) = 0 at the minima approached by the layer. The positivity of W away from ±1 and the required growth follow directly from the sign and singularity structure of the inverted expression for each range of α. We acknowledge that the manuscript does not contain an explicit verification for a concrete value of α. In the revised version we will include a short appendix that carries out the reconstruction for α = 2, computes the resulting W explicitly, and verifies that it satisfies the double-well conditions (W(±1) = 0, W > 0 elsewhere) at the formal level. This example will illustrate the emergence of the predicted degeneracy pattern. revision: partial

Circularity Check

0 steps flagged

No circularity: reconstruction proceeds from independent assumptions on the layer

full rationale

The abstract and skeptic summary describe an inverse construction that begins with an externally prescribed transition layer (power decay at infinity) and formally recovers structural features of W via the Euler-Lagrange equation. No quoted equation or self-citation in the supplied material shows that the recovered W is defined in terms of the layer decay, that a fitted parameter is relabeled as a prediction, or that a uniqueness theorem is imported from the authors' prior work to close the loop. The claimed correspondence between decay rate and regularity is therefore not forced by construction from the input layer profile; the derivation chain remains open to external verification.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are provided in the abstract.

pith-pipeline@v0.9.0 · 5400 in / 999 out tokens · 29457 ms · 2026-05-10T18:16:57.631786+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prescribe an increasing function ϕ:R→(−1,1) ... approaching them at infinity with a given decay rate ... construct a potential V such that L_s ϕ(x) = V'(ϕ(x)) and study the regularity and structure of V in terms of the asymptotic behavior of ϕ.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

lim r→−1+ V(r)/(1+r)^{2s/α+1} = α C1^{-2s/α} / (2s+α)s ... V ∈ C^{⌊2s/α⌋,1}([−1,0])

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Abatangelo and E

[AV19] N. Abatangelo and E. Valdinoci,Getting acquainted with the fractional Laplacian, Contemporary re- search in elliptic PDEs and related topics, Springer INdAM Ser., vol. 33, Springer, Cham, 2019, pp. 1–

work page 2019
[2]

Cabr´ e and E

MR3469920 [CC10] X. Cabr´ e and E. Cinti,Energy estimates and 1-D symmetry for nonlinear equations involving the half- Laplacian, Discrete Contin. Dyn. Syst.28(2010), no. 3, 1179–1206, DOI 10.3934/dcds.2010.28.1179. MR2644786 [CC14] ,Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differen- tial Equations49(2014), n...

work page doi:10.3934/dcds.2010.28.1179 2010
[3]

Monneau and S

MR371203 [MP12] R. Monneau and S. Patrizi,Homogenization of the Peierls-Nabarro model for dislocation dynamics, J. Differential Equations253(2012), no. 7, 2064–2105, DOI 10.1016/j.jde.2012.06.019. MR2946964 [PSV13] G. Palatucci, O. Savin, and E. Valdinoci,Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl...

work page doi:10.1016/j.jde.2012.06.019 2012

[1] [1]

Abatangelo and E

[AV19] N. Abatangelo and E. Valdinoci,Getting acquainted with the fractional Laplacian, Contemporary re- search in elliptic PDEs and related topics, Springer INdAM Ser., vol. 33, Springer, Cham, 2019, pp. 1–

work page 2019

[2] [2]

Cabr´ e and E

MR3469920 [CC10] X. Cabr´ e and E. Cinti,Energy estimates and 1-D symmetry for nonlinear equations involving the half- Laplacian, Discrete Contin. Dyn. Syst.28(2010), no. 3, 1179–1206, DOI 10.3934/dcds.2010.28.1179. MR2644786 [CC14] ,Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differen- tial Equations49(2014), n...

work page doi:10.3934/dcds.2010.28.1179 2010

[3] [3]

Monneau and S

MR371203 [MP12] R. Monneau and S. Patrizi,Homogenization of the Peierls-Nabarro model for dislocation dynamics, J. Differential Equations253(2012), no. 7, 2064–2105, DOI 10.1016/j.jde.2012.06.019. MR2946964 [PSV13] G. Palatucci, O. Savin, and E. Valdinoci,Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl...

work page doi:10.1016/j.jde.2012.06.019 2012