{"id":"openai-math/De-Giorgis-conjecture-in-dimension-eight","pdf":"https://github.com/openai/math/blob/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf","url":"https://github.com/openai/math/tree/main/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026","tags":["pde","elliptic-equations","allen-cahn"],"score":0,"title":"A positive resolution of De Giorgi's conjecture in dimension eight","domain":"Partial differential equations","family":"375","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.AP","summary":"We resolve De Giorgi's conjecture positively in dimension eight: every entire C 2 solution $`u:\\mathbb R^8\\to(-1,1)`$ of $`\\Delta u=u^3-u`$ with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution $`v:\\mathbb R^7\\to[-1,1]`$ of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-26","reactions":{},"formalized":false,"family_title":"De Giorgi's conjecture in dimension eight"}