Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces.

By Manuel del Pino, M. Kowalczyk, Jun-cheng Wei. 2009

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We review some recent results on construction of entire solutions to the classical semilinear elliptic equation ∆u+u−u = 0 in R . In various cases, large dilations of an embedded, complete minimal surface approximate the transition set of a solution that connects the equilibria ±1. In particular, our construction answers negatively a celebrated conjecture by E. De Giorgi in dimensions N ≥ 9.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7a77dc99b5742675f1797ec040ed222e5995a488