On De Giorgi's conjecture in dimension N ≥ 9

• 成果类型: Article
• 署名作者: del Pino, Manuel; Kowalczyk, Michal; Wei, Juncheng
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.3.3
• 发表日期: 2011
• 页码: 1485-1569
• 关键词: mean-curvature hypersurfaces; allen-cahn equation; phase-transitions; de-giorgi; singular perturbation; elliptic-equations; gradient theory; CONVERGENCE; REGULARITY; symmetry

摘要

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation Delta u + (1 - u(2))u = 0 in R-N with partial derivative(yN)u > 0 must be such that its level sets {u = lambda} are all hyperplanes, at least for dimension N <= 8. A counterexample for N >= 9 has long been believed to exist. Starting from a minimal graph F which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R-N, N >= 9, we prove that for any small alpha > 0 there is a bounded solution u(alpha)(y) with partial derivative(yN)u(alpha) > 0, which resembles tanh (t/root 2), where t = t(y) denotes a choice of signed distance to the blown-up minimal graph Gamma alpha := alpha(-1)Gamma. This solution is a counterexample to De Giorgi's conjecture for N >= 9.