Analysis of PDEs · Allen–Cahn equation · geometric phase transitions

De Giorgi Conjecture in Dimensions 5–8

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A bounded solution of the Allen–Cahn equation rises strictly in one direction. Must all of its level surfaces be parallel hyperplanes, making the solution a translated and rotated one-dimensional tanh front? The question remains open here in dimensions 5 through 8.

- Δ u = u - u^{3}, \partial_{x_{d}} u > 0.

Known results and sources

The De Giorgi conjecture asks whether every bounded strictly monotone Allen–Cahn transition in dimensions 5–8 must be a single planar tanh front; the unrestricted question remains open.

Research problem

Exact mathematical statement

The unresolved question is the following. Let $5\le d\le 8$ , and let $u: \mathbb R^d\to(-1,1)$ solve

-\Delta u=u-u^3,\qquad \partial_{x_d}u>0.

Must there be $e\in S^{d-1}$ with $e_d>0$ and $c\in\mathbb R$ such that

u(x)=\operatorname{tanh}\left(\frac{e\cdot x-c}{\sqrt2}\right)?

Equivalently, must every level set be a hyperplane? The retained source explicitly says it is not a complete proof. Its derived identities and conditional reductions remain source-reported pending independent mathematical review, and several imported framework and bridge theorems still require exact source verification.

Problem infographic

Problem at a glance

Problem-first Allen–Cahn infographic showing the equation and strict monotonicity, the conjectured planar tanh front, unrestricted dimensions 5–8 marked open, counterexamples from dimension 9 upward, and Savin's conditional theorem through dimension 8 under the uniform pure-phase end-limit hypothesis u(x', x_d) → ±1 as x_d → ±∞. — The exact target is unrestricted monotone rigidity in dimensions 5–8. Uniform pure-phase end limits give a known theorem through dimension 8, while nonplanar counterexamples begin in dimension 9; neither fact settles the retained target.

Current mathematical picture

Where work on De Giorgi Conjecture in Dimensions 5–8 stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: The source retires the total-spread Toda scalarization because it produces the wrong differential-inequality direction, rejects abstract Schrödinger stability as a replacement for nonlinear phase structure, and shows that naive linear-phase or physical-time separators give volume-order or tautological bounds rather than the required surface-order estimate. No current route bypasses endpoint control, bounded-density stable classification, and the final monotone bridge. In particular, the current work still needs a surface-order low-flux separator or equivalent endpoint argument; scale-uniform integrability and exact density-loss control on the bad-cell complement; and a cited or proved theorem turning constant or planar end states into one-dimensionality of the original monotone solution.

Route status · Narrowed route

Main reductionEndpoint density is a separate requirement

The source warns that a stable classification theorem in four dimensions does not settle original dimension five because bounded energy density of the end states is not automatic.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeObtain surface-order endpoint energy or a valid endpoint-rigidity bypass.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

De Giorgi Conjecture in Dimensions 5–8 in numbers

1.1kretained lines of mathematical investigation1,124 in the current working snapshot

Argument development: 949 · 84%
Explored or eliminated routes: 18 · 2%
Open obligations: 70 · 6%
Definitions and setup: 87 · 8%

5selected mapped statements1routes investigated4open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byEnnio De GiorgiProposed1978Status checkedAug 13, 2026

Source recordedAug 13, 2026

Recommended next task

Obtain surface-order endpoint energy or a valid endpoint-rigidity bypass.

Suggested move: Derive and audit the Euler–Lagrange or dual formulation of the graph-cut functional, then prove a competitor with action O(R^(d-2)); stop if the construction only gives volume order.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

10 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks

Obtain surface-order endpoint energy or a valid endpoint-rigidity bypass.Suggested move: Derive and audit the Euler–Lagrange or dual formulation of the graph-cut functional, then prove a competitor with action O(R^(d-2)); stop if the construction only gives volume order.

Ready to work on

Close the perforated stable-classification transfer without losing height excess on bad cells.Suggested move: Prove a scale-uniform L^(1+epsilon) estimate or a hole-filling inequality for the projected height-excess density and match its normalization and density-loss rate to the exact critical-sequence theorem.

Ready to work on

Unrestricted dimensions 5–8 target remains open

The governing source states the exact monotone Allen–Cahn question and explicitly says the present state is not a complete proof.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.

Ready to work on

Establish the final monotone bridge for every classified endpoint configuration.Suggested move: Locate or prove exact constant/planar and parallel-planar endpoint theorems, recording uniform-convergence, local-minimality, density, and dimension hypotheses rather than citing a nearby result.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026

Current statusOpen conjecture

The unrestricted monotone Allen–Cahn rigidity conjecture remains open in ambient dimensions 5 through 8. This current-status statement is an inference from the retained sources: Savin proves the result through dimension 8 only with uniform limits to the two pure phases, counterexamples are known from dimension 9 upward, and current stable and finite-index work still treats the unrestricted higher-dimensional rigidity program as conjectural. A 2025 preprint claiming the unrestricted theorem was withdrawn after its author reported a serious flaw, so it is not evidence of resolution.

[2][3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025
PreprintFlorit-Simon and Serra proved that stable Allen–Cahn solutions in R^4 with bounded energy density are one-dimensional, a relevant base classification result rather than a solution of the unrestricted dimensions 5–8 target.[5]
2025
PreprintGazoulis posted a preprint claiming the original conjecture without Savin's limiting hypothesis; arXiv records the paper as withdrawn in October 2025 because the author reported a serious flaw in the proof.[4]
2011
Peer reviewedDel Pino, Kowalczyk, and Wei constructed bounded strictly monotone nonplanar solutions in every ambient dimension at least 9.[3]
2009
Peer reviewedSavin proved hyperplane level sets in dimensions at most 8 under the additional assumption of uniform convergence to minus and plus one along the monotone coordinate.[2]

Mathematical neighborhood

Related results and reusable starting points

Current focusDe Giorgi conjecture

Solved special caseUniform-end-limit De Giorgi theorem

Uniform convergence to the two pure phases along the monotone coordinate closes the conjecture through dimension 8; the extra endpoint hypothesis is not part of the unrestricted target.

[2]

Related problemWithdrawn unrestricted proof claim

The withdrawn preprint claimed a proof of the unrestricted original statement, but the author's arXiv withdrawal reports a serious flaw; the claim has no positive evidentiary role in this workspace.

[4]

Related problemStable bounded-density Allen–Cahn classification

Stable or finite-index classification with bounded energy density supplies neighboring rigidity statements and potential inputs to reduction programs, but it is logically distinct from the unrestricted monotone conjecture.

[5][6]

Stronger or generalized formHigher-dimensional monotone rigidity

The unrestricted monotone statement extended to ambient dimensions at least 9 is false by explicit nonplanar solutions modeled on nonflat minimal graphs.

[3]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetFormal analytic infrastructure for bounded entire semilinear elliptic solutions on Euclidean space and strict monotonicity in one coordinate.
Formalization targetA formal treatment of the Allen–Cahn energy, stability quadratic form, Modica estimate, and asymptotic end states.
Formalization targetFormal geometric-measure-theory infrastructure for flatness improvement, minimal hypersurface limits, and dimension-dependent Bernstein theorems.
Formalization targetA formal proof of the exact monotone-to-one-dimensional bridge under the hypotheses used by any proposed reduction.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

3 standing statements2 proposed statements4 open questions1 narrowed routes

Statements by mathematical role5 selected mapped statements

theorem candidate1 of 51
reduction2 of 52
lemma2 of 52

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.18 displayed rows · 1 route included

retained route statementDe Giorgi Conjecture in Dimensions 5–8
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementEndpoint density is a separate requirementintermediate
retained route statementSource reports exact stable Toda-edge exclusionintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSource-reported limitationreported failure
Research targetObtain surface-order endpoint energy or a valid endpoint-rigidity bypass.open
Research targetClose the perforated stable-classification transfer without losing height excess on bad cells.open
Research targetEstablish the final monotone bridge for every classified endpoint configuration.open
Research targetUnrestricted dimensions 5–8 target remains openopen
Research targetSurface-order low-flux cut is opensuperseded
Research targetBad-cell height excess needs uniform integrabilitysuperseded
Research targetFinal monotone bridge is not suppliedsuperseded
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: The source retires the total-spread Toda scalarization because it produces the wrong differential-inequality direction, rejects abstract Schrödinger stability as a replacement for nonlinear phase structure, and shows that naive linear-phase or physical-time separators give volume-order or tautological bounds rather than the required surface-order estimate. No current route bypasses endpoint control, bounded-density stable classification, and the final monotone bridge. In particular, the current work still needs a surface-order low-flux separator or equivalent endpoint argument; scale-uniform integrability and exact density-loss control on the bad-cell complement; and a cited or proved theorem turning constant or planar end states into one-dimensionality of the original monotone solution.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeObtain surface-order endpoint energy or a valid endpoint-rigidity bypass.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointObtain surface-order endpoint energy or a valid endpoint-rigidity bypass.

De Giorgi Conjecture in Dimensions 5–8 · ready to start

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Sources and references6 cited works · next context review by Nov 13, 2026

The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.

1
On De Giorgi's conjecture in dimensions 4 and 5peer reviewed result · Nassif Ghoussoub, Changfeng Gui · Annals of Mathematics · 2003 · DOI 10.4007/annals.2003.157.313 · accessed Aug 13, 2026
2
Regularity of flat level sets in phase transitionspeer reviewed result · Ovidiu Savin · Annals of Mathematics · 2009 · DOI 10.4007/annals.2009.169.41 · MR MR2480601 · accessed Aug 13, 2026
3
On De Giorgi's conjecture in dimension N >= 9peer reviewed result · Manuel del Pino, Michał Kowalczyk, Juncheng Wei · Annals of Mathematics · 2011 · DOI 10.4007/annals.2011.174.3.3 · MR MR2846486 · accessed Aug 13, 2026
4
Minimality of level sets in phase transitionspreprint · Dimitrios Gazoulis · arXiv · 2025; withdrawn 2025-10-01 · ARXIV 2503.02604 · DOI 10.48550/arXiv.2503.02604 · accessed Aug 13, 2026
5
On stable solutions to the Allen-Cahn equation with bounded energy density in R^4preprint · Enric Florit-Simon, Joaquim Serra · arXiv · 2025 · ARXIV 2509.02739 · DOI 10.48550/arXiv.2509.02739 · accessed Aug 13, 2026
6
Phase transitions with bounded index: Parallels to De Giorgi's conjecturepreprint · Enric Florit-Simon · arXiv · 2026 · ARXIV 2602.03136 · DOI 10.48550/arXiv.2602.03136 · accessed Aug 13, 2026

Important qualifications

The review was bounded to representative primary journal pages and author-deposited preprints; it is not an exhaustive bibliography of conditional, fractional, free-boundary, stable-solution, or minimal-surface variants.
The open status in dimensions 5 through 8 is an explicitly labeled inference from the retained sources, not a claim that bibliographic search proves the nonexistence of later work.
The withdrawn arXiv:2503.02604 remains in the current research map only to document that its unrestricted proof claim was withdrawn after the author reported a serious flaw; it supplies no positive proof evidence.
A bounded search did not establish an end-to-end formalization or proof certificate. This does not establish that none exists.
The exact 1978 proceedings text in which De Giorgi posed the conjecture was not recorded as a directly reviewed source; proposer and year are therefore supported here by later peer-reviewed accounts.

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