Short··0:20·Partial differential equations
De Giorgi's Conjecture in Dimension 8: Must the Wall Be Flat?
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About this video
De Giorgi's conjecture asks whether every solution of the Allen–Cahn equation Δu = u³ − u that rises in one direction is really a flat wall, a planar tanh profile, in dimensions up to 8. It was proved in 2D (1998) and 3D (2000), known up to 8D only with an extra limit condition (Savin), and false from dimension 9 (2011). OpenAI claims a proof in dimension 8 with no extra condition, together with a classification of stable solutions in dimension 7; this is OpenAI's preprint, not yet peer reviewed.
Paper (OpenAI, catalogue result 375):
- A positive resolution of De Giorgi's conjecture in dimension eight: github.com/openai/math/blob/main/preprints/De-Giorgis-conject…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/375
About the problem
Partial differential equations·OpenAI Math·Result 375
De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:ℝ8 → (-1,1) of Δ u=u3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:ℝ7 → [-1,1] as constant wells or planar transitions, without an energy-growth assumption.
Topics
De Giorgi conjectureDe Giorgi's conjectureDe Giorgi conjecture dimension 8Allen-Cahn equationphase transitionsminimal surfacesBernstein problemmonotone solutionsstable solutionspartial differential equationsPDEelliptic PDEtanh profilecalculus of variationsgeometric analysisOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
