A counterexample to a conjecture by De Giorgi in large dimensions

@article{Pino2008ACT,
  title={A counterexample to a conjecture by De Giorgi in large dimensions},
  author={M. A. Pino and M. Kowalczyk and J. Wei},
  journal={Comptes Rendus Mathematique},
  year={2008},
  volume={346},
  pages={1261-1266}
}
  • M. A. Pino, M. Kowalczyk, J. Wei
  • Published 2008
  • Mathematics
  • Comptes Rendus Mathematique
  • We consider the Allen–Cahn equation Δu+u(1−u2)=0in RN. A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu>0, then the level sets {u=λ}, λ∈R, must be hyperplanes at least if N⩽8. We construct a family of solutions which shows that this statement does not hold true for N⩾9. To cite this article: M. del Pino et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008). 
    31 Citations
    MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2
    • 78
    • PDF
    The role of minimal surfaces in the study of the Allen-Cahn equation.
    • 24
    • Highly Influenced
    • PDF
    Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones
    • 37
    • Highly Influenced
    • PDF
    Multiplicity one and strictly stable Allen-Cahn minimal hypersurfaces
    • 7
    • PDF
    Symmetry properties of stable solutions of semilinear elliptic equations in unbounded domains
    • 1
    • PDF

    References

    SHOWING 1-10 OF 39 REFERENCES
    Towards a counter-example to a conjecture of De Giorgi in high dimensions
    • 57
    Regularity of flat level sets in phase transitions
    • 208
    • PDF
    One-dimensional symmetry of bounded entire solutions of some elliptic equations
    • 135
    • PDF
    On De Giorgi Conjecture in Dimension $N \geq 9$
    • 14
    • PDF
    The toda system and multiple-end solutions of autonomous planar elliptic problems
    • 53
    • PDF
    Phase transitions: Uniform regularity of the intermediate layers
    • 37
    • PDF