## Abstract

We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u - u^{3} = 0. In particular, we consider a minimal surface Γ in ℝ^{9} which is the graph of a nonlinear entire function x_{9} = F(x_{1},. . .,x_{8}), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation in ℝ^{9} which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and in eternal solutions to the flow by mean curvature for graphs.

Original language | English |
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Pages (from-to) | 39-65 |

Number of pages | 27 |

Journal | Milan Journal of Mathematics |

Volume | 79 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jun 2011 |

## Keywords

- Infinite dimensional Lyapunov-Schmidt reduction
- Jacobi operator
- Minimal surfaces

## ASJC Scopus subject areas

- Mathematics(all)