Solutions to the Allen Cahn Equation and Minimal Surfaces

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 \({\mathbb {R}^9}\) 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 \({\mathbb {R}^9}\) 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.