On De Giorgi's conjecture in dimension N ≥ 9

Manuel del Pino, MichaŁ Kowalczyk, Juncheng Wei

Research output: Contribution to journalArticle

112 Citations (Scopus)

Abstract

A celebrated conjecture due to De Giorgi states that any bounded so-lution of the equation Δu + (1-u2)u = 0 in R{double-struck}N with ∂yN u > 0 must be such that its level sets {u=λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Start-ing from a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R{double-struck}N, N ≥ 9, we prove that for any small α > 0 there is a bounded solution uα(y) with ∂yN uα > 0, which resembles tanh(t/√2(, where t = t(y) denotes a choice of signed distance to the blown-up mini-mal graph Γα := α-1Γ. This solution is a counterexample to De Giorgi's conjecture for N ≥ 9.

Original languageEnglish
Pages (from-to)1485-1569
Number of pages85
JournalAnnals of Mathematics
Volume174
Issue number3
DOIs
Publication statusPublished - 1 Nov 2011

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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