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. 2012 May 1;109(18):6845-50.
doi: 10.1073/pnas.1202687109. Epub 2012 Apr 12.

On De Giorgi's conjecture and beyond

Manuel Del Pino  1 , Michal Kowalczyk, Juncheng Wei
Affiliations

On De Giorgi's conjecture and beyond

Manuel Del Pino et al. Proc Natl Acad Sci U S A. .

Abstract

We consider the problem of existence of entire solutions to the Allen-Cahn equation Δu + u - u(3) = 0 in , usually regarded as a prototype for the modeling of phase transition phenomena. In particular, exploiting the link between the Allen-Cahn equation and minimal surface theory in dimensions N ≥ 9, we find a solution, u, with ∂(x(N))u > 0, such that its level sets are close to a nonplanar, minimal, entire graph. This counterexample provides a negative answer to a celebrated question by Ennio de Giorgi [De Giorgi E (1979) Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), 131-188, Pitagora, Bologna]. Our results suggest parallels of De Giorgi's conjecture for finite Morse index solutions in two and three dimensions and suggest a possible program of classification of all entire solutions.

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Conflict of interest statement

The authors declare no conflict of interest.

References

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