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On De Giorgi’s conjecture in dimensions 4 and 5. (English) Zbl 1165.35367

The authors develop an approach for establishing in some important cases, a conjecture made by De Giorgi more than 20 years ago.
Conjecture: Suppose that \(u\) is an entire solution of the equation
\[ \Delta u+u-u^3=0, \quad |u|=1, \qquad x=(x',x_n)\in\mathbb R^n \]
satisfying \(\frac{\partial u}{\partial x_n}>0\), \(x\in\mathbb R^n\). Then, at least for \(n\leq 8\), the level sets of \(u\) must be hyperplanes. The main message of the authors is that De Giorgi’s conjecture is true in dimensions \(n=4,5\) provided that the solutions are also assumed to satisfy an anti-symmetry condition.

MSC:

35J60 Nonlinear elliptic equations
35J15 Second-order elliptic equations
49J45 Methods involving semicontinuity and convergence; relaxation
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