Double layered solutions to the extended Fisher-Kolmogorov P.D.E.

TL;DR

This paper constructs double layered solutions for the extended Fisher-Kolmogorov PDE, providing the first examples of two-dimensional minimal solutions that are significant in phase transition models and related to the De Giorgi conjecture.

Contribution

It introduces the first known two-dimensional minimal solutions for the extended Fisher-Kolmogorov PDE under specific heteroclinic separation conditions.

Findings

Construction of double layered solutions

First examples of 2D minimal solutions for the PDE

Relevance to phase transition models and De Giorgi conjecture

Abstract

We construct double layered solutions to the extended Fisher-Kolmogorov P.D.E., under the assumption that the set of minimal heteroclinics of the corresponding O.D.E. satisfies a separation condition. The aim of our work is to provide for the extended Fisher-Kolmogorov equation, the first examples of two-dimensional minimal solutions, since these solutions play a crucial role in phase transition models, and are closely related to the De Giorgi conjecture.