One-dimensional symmetry for solutions of Allen Cahn fully nonlinear equations

TL;DR

This paper proves that bounded solutions to certain fully nonlinear Allen Cahn equations, which converge uniformly at infinity, depend only on one variable, revealing a one-dimensional symmetry under specific conditions.

Contribution

It establishes one-dimensional symmetry results for solutions of fully nonlinear Allen Cahn type equations under new assumptions on the forcing term.

Findings

Solutions depend only on one variable under given conditions.

Bounded solutions converge uniformly at infinity to their extrema.

Symmetry results extend previous understanding of Allen Cahn equations.

Abstract

This article presents some qualitative results for entire solutions of the fully nonlinear elliptic equations of Allen Cahn type . Precisely under some additional assumptions on the forcing term, if the solution is bounded and converges uniformly at infinity in a fixed direction to its extrema, then the solution depends only on one variable.