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On De Giorgi’s conjecture in dimension \(N\geq 9\). (English) Zbl 1238.35019
A celebrated conjecture due to E. De Giorgi [“Recent methods in non-linear analysis”, in: Proc. int. Meet., Rome 1978, 131–188 (1979; Zbl 0405.49001)] states that any bounded solution of the equation \(\Delta u + (1-u^2) u = 0\) in \(\mathbb{R}^N\) with \(\partial_{y_N}u >0\) must be such that its level sets \(\{u=\lambda\}\) are all hyperplanes, at least for dimension \(N\leq 8.\) A counterexample for \(N\geq 9\) has long been believed to exist.
In the very interesting paper under review, the authors propose a counterexample to De Giorgi’s conjecture for \(N\geq 9.\) Precisely, starting from a minimal graph \(\Gamma\) (which is not a hyperplane) found by E. Bombieri, E. De Giorgi and E. Giusti [Invent. Math. 7, 243–268 (1969; Zbl 0183.25901)] in \(\mathbb{R}^N\), \(N\geq 9\), it is proved that for any small \(\alpha >0\) there is a bounded solution \(u_\alpha(y)\) with \(\partial_{y_N}u_\alpha >0,\) which resembles \( \tanh \left ( \frac t{\sqrt{2}}\right ) \), where \(t=t(y)\) denotes a choice of signed distance to the blown-up minimal graph \(\Gamma_\alpha := \alpha^{-1}\Gamma.\)

MSC:
35J15 Second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35B08 Entire solutions to PDEs
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