One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in R$^N$

TL;DR

This paper establishes one-dimensional symmetry and Liouville type results for the extended fourth order Allen-Cahn equation in R^N, including proofs of conjectures, a priori bounds, and rigidity results for solutions with specific asymptotic behaviors.

Contribution

It proves an analogue of Gibbons' conjecture and extends symmetry and Liouville results to more general fourth order elliptic equations with nonlinearities.

Findings

Proved an analogue of Gibbons' conjecture for the fourth order Allen-Cahn equation.

Established Liouville type results for solutions with specific asymptotic limits.

Derived a priori bounds and symmetry results for semilinear fourth order elliptic equations.

Abstract

In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.