On De Giorgi's conjecture in dimension N≥9 论文

摘要

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation u + (1 -u 2 )u = 0 in R N with y N u > 0 must be such that its level sets {u = } 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 which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in R N , N 9, we prove that for any small > 0 there is a bounded solution u(y) with y N u > 0, which resembles tanh