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Del Pino M, Kowalczyk M, Wei J. On De Giorgi's conjecture and beyond. Proc Natl Acad Sci U S A. 2012;109(18):6845-50doi: 10.1073/pnas.1202687109.
Del Pino, M., Kowalczyk, M., & Wei, J. (2012). On De Giorgi's conjecture and beyond. Proceedings of the National Academy of Sciences of the United States of America, 109(18), 6845-50. https://doi.org/10.1073/pnas.1202687109
Del Pino, Manuel, et al. "On De Giorgi's conjecture and beyond." Proceedings of the National Academy of Sciences of the United States of America vol. 109,18 (2012): 6845-50. doi: https://doi.org/10.1073/pnas.1202687109
Del Pino M, Kowalczyk M, Wei J. On De Giorgi's conjecture and beyond. Proc Natl Acad Sci U S A. 2012 May 01;109(18):6845-50. doi: 10.1073/pnas.1202687109. Epub 2012 Apr 12. PMID: 22499785; PMCID: PMC3344995.
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Proc Natl Acad Sci U S A. 2012 May 01;109(18):6845-50. doi: 10.1073/pnas.1202687109. Epub 2012 Apr 12.

On De Giorgi's conjecture and beyond.

Proceedings of the National Academy of Sciences of the United States of America

Manuel Del Pino, Michal Kowalczyk, Juncheng Wei

Affiliations

  1. Departamento de Ingeniería Matemática and Center for Mathematical Modeling, Universidad de Chile, Casilla 170/3, Santiago, Chile.

PMID: 22499785 PMCID: PMC3344995 DOI: 10.1073/pnas.1202687109

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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