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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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