On a conjecture of De Giorgi

On a conjecture of De Giorgi

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Ovidiu Savin, Columbia University
Fine Hall 314

In 1978 De Giorgi made a conjecture about the symmetry of global solutions to a certain semilinear elliptic equation. He stated that monotone, bounded solutions of $$ \triangle u=u^3-u$$ in $\mathbb{R}^n$ are one dimensional (i.e. the level sets of $u$ are hyperplanes) at least in dimension $n \le 8$. This problem is in fact closely related to the theory of minimal surfaces and it is sometimes referred to as "the $\varepsilon$ version of the Bernstein problem for mininimal graphs". In my talk I will explain this relation and I will give an idea about the proof of this conjecture for $n \le 8$. We mention that recently Del Pino, Kowalzyk and Wei provided a counterexample in dimension $n \ge 9$.