On De Giorgi ' s Conje ture in Dimensions
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منابع مشابه
A Counterexample to a Conjecture by De Giorgi in Large Dimensions
We consider the Allen-Cahn equation ∆u+ u(1− u) = 0 in R . A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu > 0, then the level sets {u = λ}, λ ∈ R, must be hyperplanes at least if N ≤ 8. We construct a family of solutions which shows that this statement does not hold true for N ≥ 9. Résumé. Un contre-exemple à la conjecture de...
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A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u2)u = 0 in RN 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. Based on a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in RN , N ≥ 9, we prove t...
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تاریخ انتشار 2001